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Multiscale modeling

Multiscale modeling is a computational framework that integrates simulations across multiple spatial, temporal, and physical scales to analyze complex systems in science and engineering, linking detailed microscopic processes—such as atomic interactions—to emergent macroscopic behaviors that single-scale approaches cannot adequately capture due to computational limitations or loss of resolution.^[1] This methodology employs a hierarchy of models, ranging from quantum mechanics and molecular dynamics at fine scales to continuum mechanics like finite element methods at coarser scales, enabling efficient predictions of system properties while balancing accuracy and cost.^[2] Central to multiscale modeling are two primary strategies: sequential approaches, which hierarchically coarse-grain information from finer to coarser scales using techniques like the Cauchy-Born rule or free-energy calculations, and concurrent methods, which dynamically couple models across domains to resolve local phenomena without relying on phenomenological parameters.^[1] Challenges in implementation include seamless scale coupling to avoid artifacts, handling statistical fluctuations and memory effects, and bridging vast timescale disparities—from femtoseconds in atomic vibrations to seconds in structural responses—often addressed through algorithms like the Heterogeneous Multiscale Method (HMM) or equation-free schemes.^[2] These principles allow for error-controlled simulations that exploit scale separation, such as in perturbation analysis or renormalization group theory, to derive effective equations for multiscale dynamics.^[1] The approach finds extensive applications in materials science, where it simulates dislocation motion, phase transformations, and fracture to design advanced alloys and composites; in biomechanics, linking organ-level loading to cellular deformations for injury prediction; and in fluid mechanics, modeling nanoscale flows in porous media or turbulent phenomena via hybrid continuum-molecular methods.^[2] In biomedicine, it integrates multiphysics data to elucidate disease mechanisms, such as in cardiovascular systems, while in environmental engineering, it aids in simulating pollutant transport across scales.^[3] Overall, multiscale modeling drives innovation by providing mechanistic insights into phenomena like nanotechnology device performance and sustainable energy materials, with ongoing advancements incorporating machine learning for enhanced scalability and predictive power.^[4]

Fundamentals

Definition and Principles

Multiscale modeling involves the development and integration of mathematical and computational models to capture system behaviors across disparate spatial and temporal scales, from atomic to macroscopic levels, enabling the prediction of overall properties without simulating every fine detail. This approach bridges microscopic accuracy with macroscopic efficiency, addressing complex phenomena in fields like physics, materials science, and biology by linking models that operate at different resolutions.^[1]^[5] Central principles include the separation of scales, which categorizes phenomena by spatial extents—atomic around $10^{-10} m, mesoscale from $10^{-9} to $10^{-6} m, and macroscale beyond $10^{-6} m—and temporal spans from femtoseconds ($10^{-15} s) to seconds. This separation exploits the hierarchical nature of physical laws, as seen in the Navier-Stokes equations for macroscale fluid dynamics, which emerge from underlying molecular interactions without resolving them explicitly. Information flows bidirectionally: upscaling aggregates fine-scale data, such as deriving effective parameters like stress tensors from atomic simulations, to inform coarse models; downscaling refines macroscale solutions, for instance by interpolating velocity fields to guide microscopic refinements.^[1]^[6] Fundamentally, fine-scale dynamics are often described by the master equation for probabilistic state transitions:

\frac{\partial u}{\partial t} = \sum_j a_j(\mathbf{x}) \bigl( u(\mathbf{x} + \boldsymbol{\nu}_j, t) - u(\mathbf{x}, t) \bigr),

where u is the probability density, a_j are transition rates, and \boldsymbol{\nu}_j are jumps, which connects to continuum descriptions via the Liouville equation governing phase-space evolution in classical mechanics. Single-scale models fail in complex systems because they overlook emergent properties—like turbulence in fluids, where macroscopic chaos arises from unresolved molecular collisions—leading to inaccurate empirical relations and prohibitive computational costs across scale disparities.^[1]^[5]

Scale Hierarchies and Coupling

In multiscale modeling, physical systems are structured into hierarchies of scales that reflect the natural organization of phenomena across lengths and times, typically progressing from quantum and atomic levels to molecular or mesoscopic intermediates and finally to continuum or macroscopic descriptions. At the quantum/atomic scale, interactions occur over angstroms and femtoseconds, governing electronic structures and interatomic forces, as seen in solid materials where electron configurations determine bonding in crystals.^[7] The molecular/mesoscopic scale bridges this by aggregating atoms into larger entities like polymers or grains, spanning nanometers and picoseconds to microseconds, where collective behaviors such as diffusion or phase transitions emerge.^[1] At the continuum/macroscopic scale, meters and seconds dominate, capturing bulk properties like stress-strain responses through partial differential equations. This hierarchy enables systematic analysis by exploiting scale separation, where finer details inform coarser behaviors without resolving every atomic motion.^[1] Coupling mechanisms facilitate information transfer across these hierarchies to maintain model consistency. Upscaling aggregates fine-scale data into effective coarse-scale parameters, for example, by averaging atomic simulation outputs to compute macroscopic elastic moduli that represent homogenized material stiffness.^[1] Downscaling, conversely, imposes coarse-scale constraints—such as imposed strains or velocities—onto finer models to guide local simulations while preserving global consistency.^[1] These transfers occur either sequentially, where parameters are precomputed at finer scales and passed upward before coarse simulations proceed, or concurrently, where scales are simulated simultaneously with real-time handshaking at interfaces to capture dynamic interactions.^[1] Sequential approaches reduce computational cost but may overlook transient couplings, while concurrent methods enhance accuracy at the expense of complexity.^[7] A cornerstone of coupling in periodic media is homogenization theory, which derives effective macroscopic properties by asymptotically expanding the solution over multiple scales. The basic formulation assumes a fast variable y = x / \varepsilon (where \varepsilon is the small periodicity parameter) and expands the solution as

u^\varepsilon(x) = u_0(x, y) + \varepsilon u_1(x, y) + \varepsilon^2 u_2(x, y) + \cdots,

leading to cell problems on the unit periodic domain that yield homogenized coefficients, such as effective conductivity or elasticity tensors, through volume averaging.^[8] Error control in coupling relies on interface conditions, like flux continuity or displacement matching, to minimize discrepancies between scales and ensure numerical stability.^[1] Despite these advances, coupling poses significant challenges, including information loss during upscaling, where fine-scale heterogeneities are averaged out, potentially overlooking critical fluctuations or rare events that influence macroscopic behavior.^[9] Bidirectional flows exacerbate this by propagating errors across scales, leading to instabilities such as artificial reflections at interfaces or divergence in concurrent simulations.^[10] These issues demand robust error estimators and adaptive strategies to bound inaccuracies without excessive computation.^[1]

Historical Development

Early Foundations

The early foundations of multiscale modeling emerged from classical physics, where Isaac Newton's Philosophiæ Naturalis Principia Mathematica (1687) introduced the laws of motion that underpin continuum mechanics, enabling deterministic descriptions of macroscopic phenomena such as fluid flow and solid deformation. These laws treated systems at large scales as continuous media, but they inherently overlooked underlying microscopic constituents like atoms and molecules. In the 1860s, James Clerk Maxwell laid crucial groundwork with his kinetic theory of gases, positing that macroscopic properties like pressure and viscosity arise from the statistical behavior of countless colliding particles, thus initiating the conceptual bridge between molecular and continuum scales.^[11] Ludwig Boltzmann advanced this framework in the 1870s through statistical mechanics, formulating the Boltzmann equation to describe the evolution of particle distribution functions and introducing the H-theorem, which mathematically demonstrated how microscopic reversibility leads to irreversible macroscopic entropy increase, effectively linking atomic-scale dynamics to thermodynamic observables.^[11]^[12] Building on kinetic theory, Albert Einstein's 1905 analysis of Brownian motion provided empirical validation for atomic existence by modeling the erratic paths of suspended particles as resulting from collisions with fluid molecules, deriving the diffusion coefficient via the Stokes-Einstein relation and highlighting fluctuations that connect microscale randomness to macroscale transport.^[11]^[13] The early 20th century saw further scale-bridging with Erwin Schrödinger's 1926 wave equation, which governs quantum phenomena at atomic and subatomic levels, allowing for the probabilistic description of electron behavior and laying the basis for transitioning quantum effects to classical regimes in later multiscale contexts.^[11]^[14] Simultaneously, the Chapman-Enskog expansion, initiated by Sydney Chapman in the 1910s and refined by David Enskog in 1917, offered a perturbative approach to solve the Boltzmann equation asymptotically, deriving the Navier-Stokes equations for viscous fluids from kinetic theory and illustrating how transport coefficients emerge across scales.^[11]^[15] Contributions from these pioneers—Newton, Maxwell, Boltzmann, Einstein, Schrödinger, Chapman, and Enskog—established analytical paradigms for scale integration, emphasizing statistical averaging and perturbation to reconcile disparate physical descriptions. However, these early developments were constrained by their reliance on hand-derived solutions in a pre-computational era, often employing approximation techniques like Poincaré's averaging method from the 1890s, which simplified oscillatory perturbations in nonlinear systems such as celestial orbits but proved inadequate for highly coupled or non-perturbative multiscale interactions.^[11]^[16]

Key Milestones

In the mid-20th century, the development of Monte Carlo methods provided a foundational tool for sampling fine-scale phenomena in complex systems, enabling statistical simulations of atomic and molecular behaviors through random sampling techniques.^[17] Concurrently, finite element methods emerged in the 1950s as a key approach for macroscale simulations, particularly in structural engineering, by discretizing continuous domains into finite elements to solve partial differential equations for stress and deformation. These innovations, driven by early computational capabilities, laid the groundwork for bridging microscopic and macroscopic scales in engineering and physics applications. During the 1970s and 1990s, molecular dynamics simulations matured significantly, building on pioneering work in the 1950s and 1960s that demonstrated the feasibility of numerically integrating equations of motion for hundreds of interacting particles to study liquid and gas dynamics.^[18] Similarly, density functional theory advanced from its theoretical formulation in 1964, which established that the ground-state properties of interacting electron systems are uniquely determined by the electron density, to practical implementations in the 1980s that enabled efficient quantum mechanical calculations for materials and chemical systems.^[19] From the 2000s onward, hybrid quantum mechanics/molecular mechanics (QM/MM) approaches gained widespread adoption, originating from a 1976 study that combined quantum calculations for reactive regions with classical mechanics for surrounding environments to model enzymatic reactions realistically.^[20] Coarse-graining frameworks, such as the MARTINI model introduced in 2004, further accelerated simulations by mapping atomic details to larger beads, facilitating studies of lipid membranes and biomolecular assemblies at mesoscales.^[21] The establishment of the SIAM Journal on Multiscale Modeling & Simulation in 2003 reflected the field's growing maturity, providing a dedicated venue for interdisciplinary research on multiscale algorithms and applications.^[22] In recent years up to 2025, exascale computing initiatives by the U.S. Department of Energy have enabled unprecedented multiscale simulations, with systems like Aurora deployed in 2025 to support high-fidelity modeling across scales in energy and national security applications.^[23]^[24] Additionally, the integration of machine learning, exemplified by physics-informed neural networks introduced in 2017, has enhanced multiscale modeling by embedding physical laws directly into neural network training to solve partial differential equations efficiently from data.^[25]

Modeling Approaches

Hierarchical Methods

Hierarchical methods in multiscale modeling involve sequential integration of models across scales, where simulations at one level parameterize or inform models at another level without simultaneous execution, enabling efficient bridging of disparate length and time scales. These approaches typically proceed either bottom-up, aggregating fine-scale details into effective coarse-scale descriptions, or top-down, applying macroscopic constraints to guide microscale dynamics. By decoupling scales, hierarchical methods facilitate the transfer of information unidirectionally, reducing the need for full-resolution simulations across the entire domain while preserving key physical behaviors. In bottom-up hierarchical modeling, fine-scale simulations, such as molecular dynamics (MD), generate parameters for coarser continuum models, allowing atomic-level phenomena to inform macroscopic properties like transport coefficients. For instance, MD trajectories can compute viscosity via the Green-Kubo formula, where the shear viscosity \mu is obtained from the autocorrelation of the off-diagonal stress tensor components: \mu = \frac{V}{k_B T} \int_0^\infty \langle \sigma_{xy}(t) \sigma_{xy}(0) \rangle dt, with V the system volume, k_B Boltzmann's constant, and T temperature; this viscosity is then directly inserted into Navier-Stokes equations for fluid flow simulations. Another representative example is extracting elastic constants from atomic simulations of solids using Green's function molecular dynamics (GFMD), which leverages the fluctuation-dissipation theorem to relate thermal fluctuations in atomic displacements to the elastic Green's function G_{\alpha\beta}(\mathbf{q}) in reciprocal space. The inverse correlation matrix, scaled by thermal energy, yields stiffness coefficients \Phi_{\alpha\beta}(\mathbf{q}), enabling the computation of effective moduli for continuum elasticity models while simulating only surface atoms to achieve near-linear scaling in system size. Top-down hierarchical approaches impose constraints from coarser scales onto finer models to focus sampling on relevant regions of phase space, enhancing efficiency in exploring targeted configurations. A key technique is constrained MD, where macroscopic variables—such as collective coordinates from a coarse-grained (CG) model—are used to apply time-dependent restraints on atomistic simulations, guiding the system toward desired states like protein conformational transitions. For example, in multiscale enhanced sampling, a variational autoencoder latent space derived from CG distances generates interpolated restraints for atomistic MD, refining ensembles from single-basin trajectories into multi-basin distributions with high exchange acceptance rates in replica-exchange schemes. Central techniques in hierarchical methods include coarse-graining, which systematically reduces degrees of freedom by mapping atomistic details to effective CG interactions, and renormalization group (RG) methods for capturing scale-invariant behaviors near critical points. In coarse-graining for polymers, iterative Boltzmann inversion (IBI) refines CG potentials by iteratively matching target radial distribution functions from reference MD simulations, starting with an initial guess and updating via V_{CG}^{(n+1)}(r) = V_{CG}^{(n)}(r) - k_B T \ln \left[ g_{target}(r) / g_{CG}^{(n)}(r) \right], where g denotes pair correlations; this is often combined with force-matching, which minimizes the least-squares difference between all-atom and CG forces to derive bonded and non-bonded terms. For critical phenomena, RG methods employ block-spin transformations or flow equations to coarse-grain Hamiltonians, revealing universal scaling through the β-function, defined as \beta(g) = \frac{dg}{dl} where l = \ln(\Lambda / k) is the RG flow parameter and g a coupling constant; fixed points where \beta(g^*) = 0 dictate critical exponents, such as the correlation length exponent \nu from the eigenvalue of the linearized flow. These methods offer significant advantages, particularly reduced computational cost compared to fully atomistic simulations, as fine-scale computations are localized and reused across coarse-scale iterations. In solid mechanics, homogenization techniques exemplify this by averaging microscale responses—such as stress-strain relations in fiber-reinforced composites—into effective macroscopic tensors via computational homogenization, where representative volume elements (RVEs) under periodic boundary conditions yield homogenized stiffness matrices with errors below 5% relative to direct measurements while cutting simulation time by orders of magnitude through parallelizable microscale boundary value problems.

Concurrent and Hybrid Methods

Concurrent approaches in multiscale modeling involve domain decomposition techniques where regions of different resolutions—such as fine-scale molecular dynamics (MD) and coarse-scale continuum models—simulate simultaneously and exchange information in real-time at their interfaces. This enables the capture of dynamic interactions across scales without sequential handoffs, allowing adaptive refinement in critical areas like interfaces or defects. For instance, adaptive mesh refinement (AMR) in computational fluid dynamics (CFD) can be coupled with MD simulations at boundaries to model phenomena such as wetting or fracture propagation, where atomic-level details influence macroscopic flow.^[26] A prominent example of concurrent methods is the Heterogeneous Multiscale Method (HMM), which couples microscale solvers (e.g., MD or Monte Carlo) to macroscale differential equations by estimating missing macroscopic data from local fine-scale simulations, enabling efficient resolution of multiscale problems like turbulent flows or wave propagation.^[2] Hybrid methods blend disparate modeling paradigms to bridge scales efficiently, often partitioning the system into regions treated by different theories. A prominent example is the quantum mechanics/molecular mechanics (QM/MM) approach, which applies quantum mechanical (QM) calculations to a reactive core (e.g., active sites in enzymes) while using molecular mechanics (MM) for the surrounding environment to reduce computational cost. The total energy is computed as E_{\text{total}} = E_{\text{QM}} + E_{\text{MM}} + E_{\text{boundary}}, where the boundary term accounts for interactions across the QM-MM interface, such as electrostatic embedding or link-atom schemes to handle covalent bonds. This method, foundational for biomolecular simulations, has been widely adopted for studying enzymatic reactions and material defects.^[27] Lattice Boltzmann methods (LBM) serve as hybrid meso-continuum bridges, simulating fluid flows at intermediate scales by evolving particle distribution functions on a lattice, which naturally couples microscopic collisions to macroscopic hydrodynamics via the Chapman-Enskog expansion. In multiscale contexts, LBM facilitates concurrent coupling with continuum solvers for problems like porous media flow or multiphase transport, enabling real-time information transfer without explicit scale separation. Advanced variants include multigrid methods for partial differential equations (PDEs), which accelerate convergence across scales using the V-cycle algorithm: starting from a fine grid, restricting residuals to coarser grids for smoothing low-frequency errors, interpolating corrections back, and iterating until convergence. This hierarchical cycling reduces the total computational work from O(N^2) to O(N), independent of grid size N, where N is the number of grid points, making it ideal for multiscale PDEs in elasticity or diffusion. Machine learning hybrids enhance these frameworks by employing neural networks as surrogate models for fine-scale physics, trained on MD datasets to predict effective coarse-scale behaviors like force fields or transport coefficients. For example, graph neural networks can learn interatomic potentials from ab initio data, enabling concurrent simulations of large systems with near-QM accuracy at MM speeds. In granular flows, handshaking regions exemplify concurrent coupling, where discrete element methods (DEM) in high-fidelity zones transition smoothly to continuum descriptions via overlapping buffers that enforce momentum and mass conservation, as seen in modeling shear bands or avalanches. These techniques contrast with sequential hierarchies by allowing bidirectional feedbacks but require careful validation to ensure interface consistency.^[4]

Applications

Materials Science

In materials science, multiscale modeling bridges atomic-scale phenomena to macroscopic properties, enabling the prediction of material behavior under various conditions such as mechanical loading and thermal processing. This approach integrates quantum mechanical calculations, like density functional theory (DFT), with classical molecular dynamics (MD) simulations to capture defect dynamics at the atomic level, which then inform mesoscale models for broader structural evolution. For instance, DFT and MD have been employed to study dislocation motion in metals, revealing how atomic-scale interactions govern plastic deformation and strength.^[28] These simulations demonstrate that dislocation velocities in body-centered cubic metals can vary by orders of magnitude with temperature and stress, providing critical parameters for higher-scale models.^[29] At the mesoscale, phase-field models simulate microstructure evolution, such as grain growth and phase transformations, by solving diffuse-interface equations that account for interfacial energies and kinetics without explicitly tracking sharp boundaries.^[30] This method has successfully predicted the coarsening of precipitates in alloys during annealing, highlighting how curvature-driven flows influence overall material homogeneity.^[31] Transitioning from meso- to macroscale, multiscale frameworks couple crystal plasticity models with finite element analysis to predict fatigue life in polycrystalline materials. Crystal plasticity finite element (CPFE) methods incorporate orientation-dependent slip systems derived from lower-scale simulations, enabling accurate forecasting of stress concentrations and damage accumulation under cyclic loading.^[32] For example, these coupled models have been applied to quantify fatigue crack initiation in titanium alloys, incorporating microstructural features like alpha-beta phases.^[33] In fiber-reinforced composites, hierarchical homogenization techniques upscale microscale damage mechanisms to simulate crack propagation, where representative volume elements (RVEs) compute effective stiffness and fracture toughness. Such approaches reveal that delamination in carbon-fiber epoxy systems initiates at fiber-matrix interfaces, propagating under mode I loading.^[34] This homogenization bridges the gap between nanoscale fiber pull-out and macroscale laminate failure, aiding in the design of damage-tolerant structures. Specific applications highlight the versatility of these methods in nanomaterials and polymers. In carbon nanotubes (CNTs), quantum mechanics/molecular mechanics (QM/MM) hybrid simulations assess interfacial strength in CNT-polymer composites, demonstrating that covalent bonding via amine groups enhances load transfer by 20-30% over van der Waals interactions.^[35] These models predict ultimate tensile strengths exceeding 50 GPa for functionalized single-walled CNTs, crucial for lightweight reinforcements.^[36] For polymers, coarse-grained MD simulations capture rheological behavior, such as viscoelastic flow in entangled chains, by mapping atomistic details to bead-spring representations that access experimentally relevant timescales.^[37] This has elucidated shear-thinning in polyethylene melts, where relaxation times scale with molecular weight as \tau \propto M^{3.4}, informing processing parameters for extrusion and molding.^[38] Multiscale modeling has accelerated materials discovery, particularly for high-entropy alloys (HEAs) in the 2010s, by combining atomistic simulations with continuum models to explore vast compositional spaces. These efforts identified refractory HEAs like MoNbTaW with superior dislocation mobilities at high temperatures, enabling creep-resistant designs for aerospace applications.^[29] Through iterative DFT-MD-dislocation dynamics workflows, multiscale approaches have explored compositional spaces for high-entropy alloys. Recent machine learning-assisted multiscale designs, as of 2024, further accelerate discovery of energy materials by integrating data-driven predictions across scales.^[39] Such advances underscore the role of multiscale approaches in tailoring microstructures for enhanced performance, from defect engineering to property optimization.^[40]

Biological and Biomedical Systems

Multiscale modeling in biological and biomedical systems integrates processes across length and time scales, from molecular interactions to organ-level behaviors, to elucidate complex phenomena such as disease progression and therapeutic responses. This approach is essential for capturing emergent properties in living systems, where stochastic events at the nanoscale influence macroscopic outcomes like tissue remodeling or immune responses. By coupling discrete simulations of individual molecules or cells with continuum descriptions of bulk transport, these models provide insights into dynamic biological processes that single-scale methods cannot resolve.^[41] At the molecular-to-cellular scale, molecular dynamics (MD) simulations enable detailed examination of protein folding and conformational changes critical for cellular function. For instance, the Anton supercomputer has facilitated millisecond-scale all-atom MD simulations of proteins, revealing folding pathways and intermediate states that were previously inaccessible due to computational limitations. These simulations, achieving timescales up to 1 millisecond for systems like the bovine pancreatic trypsin inhibitor, demonstrate how atomic-level fluctuations drive functional dynamics in biomolecules. Complementing MD, reaction-diffusion equations model intracellular signaling pathways, describing how morphogens or second messengers propagate spatial patterns through diffusion and nonlinear reactions. Such models have been applied to pathways like Wnt signaling in development, where activator-inhibitor dynamics generate graded concentrations that instruct cell fate decisions.^[42] Bridging cellular to tissue scales, agent-based models (ABM) coupled with continuum mechanics simulate collective behaviors in pathological contexts, such as tumor growth and angiogenesis. In these hybrid frameworks, individual cells are represented as discrete agents that proliferate, migrate, and interact via rules derived from mechanobiology, while nutrient and vascular factors evolve according to partial differential equations for diffusion and consumption. A notable example is the modeling of avascular tumor spheroids transitioning to vascularized states, where endothelial cells form sprouts in response to vascular endothelial growth factor (VEGF) gradients, influencing tumor invasion rates by up to 50% in simulated scenarios. These models highlight how mechanical stresses from extracellular matrix remodeling couple with biochemical cues to drive tissue-level heterogeneity.^[43] Specific applications include drug delivery systems, where Brownian dynamics tracks nanoparticle diffusion and adhesion at the cellular scale, linked to pharmacokinetic models at the organ level. For example, multiscale simulations show that smaller nanoparticles, such as 50 nm compared to 200 nm, exhibit deeper penetration into tumor interstitium due to reduced entrapment in perivascular regions.^[44] In neuroscience, MD of ion channels informs parameters for Hodgkin-Huxley-type models, which are then embedded in neural network simulations to predict network excitability. Simulations of voltage-gated sodium channels reveal gating kinetics that alter action potential propagation, scaling to explain epileptic seizure dynamics in cortical networks.^[44] The integration of multiscale modeling with machine learning has advanced personalized medicine, particularly in cancer pharmacodynamics, by optimizing treatment regimens based on patient-specific tumor evolution. Reinforcement learning informed by multiscale models has been used to predict adaptive responses to therapies like immunotherapy. Advances like neural master equations, as of 2025, enhance modeling of molecular processes in disease dynamics.^[45] These approaches, incorporating genomic data and dynamical simulations, enable tailoring of drug combinations to individual resistance profiles.

Fluid Dynamics and Environmental Systems

Multiscale modeling in fluid dynamics and environmental systems bridges microscopic interactions, such as molecular collisions in nanofluidics, to macroscopic phenomena like global atmospheric circulation. At the molecular-to-mesoscale level, molecular dynamics (MD) simulations capture nanoscale fluid behaviors, including slip at boundaries where traditional no-slip conditions fail due to molecular layering and surface interactions. For instance, MD studies reveal that slip lengths in simple fluids can vary with surface curvature, enhancing predictions of flow resistance in nanochannels. Dissipative particle dynamics (DPD) extends this by modeling mesoscale phenomena in nanofluidic systems, such as polymer-grafted channels where stimuli-responsive brushes control solvent flow through hydrodynamic interactions. These methods enable accurate representation of transport properties in confined geometries, where continuum assumptions break down. Complementing MD and DPD, the lattice Boltzmann method (LBM) simulates mesoscale flows in porous media by discretizing the Boltzmann equation on a lattice, effectively handling complex geometries like rock pores without explicit boundary tracking. LBM has been unified for multiscale porous systems, allowing seamless transitions from pore-scale velocity profiles to Darcy-scale permeability estimates, improving simulations of groundwater flow and filtration processes. Transitioning to mesoscale-to-macroscale coupling, kinetic theory-based approaches link particle-level descriptions to continuum equations for rarefied gases, where the Knudsen number exceeds continuum validity. Direct simulation Monte Carlo (DSMC) methods, rooted in kinetic theory, couple with Navier-Stokes solvers at interfaces to model flows in microdevices or high-altitude atmospheres, capturing non-equilibrium effects like velocity slip and temperature jumps. This hybrid framework resolves disparities between kinetic and hydrodynamic regimes, as demonstrated in unified gas-kinetic schemes that preserve conservation laws across scales. In global circulation models (GCMs), subgrid parameterizations represent unresolved cloud processes by statistically modeling microphysical interactions, such as droplet nucleation and radiative transfer, within coarser grid cells. The multiscale modeling framework (MMF) embeds cloud-resolving models into GCMs to explicitly simulate convective clouds, reducing parameterization errors and improving precipitation forecasts. Specific applications in atmospheric modeling integrate aerosol microphysics—from particle activation and coagulation at submicron scales—to synoptic weather prediction. The Weather Research and Forecasting model with aerosol-cloud interactions (WRF-ACI) couples bin-resolved microphysics schemes with dynamical cores, quantifying how aerosols alter cloud droplet spectra and precipitation efficiency, leading to more accurate regional weather simulations. In ocean dynamics, nested grid approaches upscale eddy-resolving simulations (resolutions ~1-10 km) to basin-scale models (~100 km), capturing submesoscale instabilities that drive heat and nutrient transport. Multi-nest primitive equation models enable two-way coupling, where fine-grid eddies feedback into large-scale currents, enhancing representations of western boundary currents like the Gulf Stream. These multiscale strategies have advanced climate forecasting, particularly in IPCC assessments from the 2010s to 2020s, by incorporating turbulence closures that parameterize subgrid-scale mixing in ocean and atmosphere components. For example, MMF-based GCMs in AR6 projections better resolve turbulent fluxes, reducing biases in sea surface temperature and cloud feedbacks, thereby refining ensemble predictions of global warming scenarios. Concurrent coupling methods at scale interfaces, though computationally intensive, further mitigate scaling challenges in these hybrid simulations.

Challenges and Future Directions

Computational and Validation Challenges

Multiscale modeling encounters significant computational challenges due to the high dimensionality inherent in coupling phenomena across disparate scales, which exacerbates the curse of dimensionality and leads to exponential increases in computational costs as the number of variables grows.^[46]^[47] For instance, simulating atomic-level details in materials science applications can require tracking millions of degrees of freedom, rendering traditional numerical methods infeasible without dimensionality reduction techniques.^[46] Parallelization is essential to manage these demands, particularly for domain coupling in concurrent methods, where frameworks like the Message Passing Interface (MPI) enable distributed computing across clusters to handle inter-scale interactions efficiently.^[48]^[49] However, effective MPI implementation requires careful load balancing to avoid bottlenecks in data exchange between fine- and coarse-scale domains, as seen in simulations of fluid-structure interactions.^[48] Exascale simulations amplify storage requirements, often generating petabytes of data from multiscale runs that capture transient behaviors over extended time periods, necessitating advanced I/O strategies to prevent I/O bottlenecks on supercomputers.^[50]^[51] Validation and verification (V&V) of multiscale models are complicated by the absence of ground truth data at intermediate scales, where experimental measurements are sparse or infeasible, making it difficult to confirm the fidelity of scale-bridging assumptions.^[52]^[53] Uncertainty propagation further hinders reliability, with methods like Monte Carlo sampling and sensitivity analysis used to quantify how microscale variabilities affect macroscale predictions, though these approaches are computationally intensive for high-dimensional inputs.^[54]^[55] Error bounds are often derived from a posteriori estimates, which provide adaptive indicators for mesh refinement in finite element-based multiscale methods by assessing residuals post-simulation.^[56]^[57] Specific issues include inconsistencies in parameter transfer between scales, where upscaling from microscale simulations to macroscale models can introduce discrepancies due to averaging assumptions that overlook local heterogeneities.^[58] Reproducibility in stochastic simulations poses another challenge, as random number generation and coupling protocols can lead to variations across runs, particularly in biological systems with inherent noise.^[59]^[60] Established V&V frameworks, such as the ASME V&V 40 standard, guide credibility assessment by integrating risk-informed processes that evaluate model relevance, verification rigor, and validation evidence tailored to application contexts like medical device simulations.^[61]^[62]

Emerging Trends

One of the most prominent emerging trends in multiscale modeling is the integration of machine learning (ML) and artificial intelligence (AI) to accelerate simulations and enhance predictive accuracy across scales. ML-assisted interatomic potentials (MLIPs), such as those based on equivariant graph neural networks like MACE and NequIP, have enabled efficient atomistic simulations that rival density functional theory (DFT) accuracy while reducing computational costs by orders of magnitude, for instance, achieving a mean absolute error of 0.18 THz in phonon dispersion predictions for energy materials.^[39] This approach facilitates high-throughput screening, as demonstrated by the GNoME platform, which discovered over 381,000 stable materials, expanding the known materials database by an order of magnitude and supporting applications in batteries and photovoltaics.^[39] Generative AI models, including variational autoencoders and diffusion models, further enable inverse design by generating novel structures, such as 11,630 new 2D materials with formation energies below 0.3 eV/atom above the convex hull.^[39] Another key development involves hybrid integrated computational materials engineering (ICME) frameworks that link atomic-scale composition to mesoscale microstructure evolution through ML and nanoscale simulations. In nickel-based superalloys, such frameworks combine CALPHAD thermodynamics, molecular dynamics (MD), and ML models like SevenNet potentials to screen billions of compositions, reducing candidates from 2 billion to 12 viable alloys with 99.3% accuracy in phase prediction, achieving a 60,000-fold efficiency gain over traditional methods.^[63] These approaches incorporate diffusion kinetics from databases like Thermo-Calc TCNI12, predicting properties such as aluminum diffusion coefficients below 1.04 × 10⁻¹⁶ m²/s, and extend to high-entropy alloys and steels for predictive microstructure design.^[63] In 2D materials, multiscale techniques integrating DFT, MD, phase-field modeling, and ML have advanced property predictions, such as graphene's thermal conductivity of 910–1655 W m⁻¹ K⁻¹ and MoS₂ bandgap reductions under 2% strain, addressing challenges in system size limitations through hybrid physics-ML surrogates.^[64] Emerging computational paradigms also emphasize adaptive and dynamic partitioning, including multiple movable quantum mechanics (QM) regions within classical or continuum environments, to model transient processes like electron transfer in photosystems with unprecedented spatiotemporal resolution.^[65] Quantum computing integration, via methods like variational quantum eigensolvers on noisy intermediate-scale quantum devices, promises to scale electronic structure calculations for complex systems, complementing ML for fault-tolerant quantum simulations.^[65] These trends promote sustainability by minimizing resource-intensive ab initio computations and fostering user-friendly interfaces through AI and virtual reality, broadening accessibility for interdisciplinary fields like biochemistry and nanoscience.^[65] Overall, such advancements position multiscale modeling as a cornerstone for autonomous material discovery and multiphysics simulations in energy, electronics, and beyond.^[65]

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[PDF] 1926
The paper gives an account of the author's work on a new form of quantum theory. §1. The Hamiltonian analogy between mechanics and optics. §2. The.
[15]
[PDF] The Mathematical Theory of Non-uniform Gases
... Chapman and T. G. Cowling when writing The Mathematical Theory of Non-uniform Gases and thus came to be known as the Chapman-Enskog method. This is a ...
[16]
Poincaré, celestial mechanics, dynamical-systems theory and “chaos”
I will show how the classical problems of celestial mechanics led Poincaré to ask fundamental questions on the qualitative behavior of differential equations.
[17]
The Monte Carlo Method - Taylor & Francis Online
The method is, essentially, a statistical approach to the study of differential equations, or more generally, of integro-differential equations.
[18]
Studies in Molecular Dynamics. I. General Method - AIP Publishing
B. J. Alder and T. E. Wainwright, Proceedings of the I.U.P.A.P. Symposium on Statistical Mechanical Theory of Transport Processes, Brussels, 1956 (Interscience ...Missing: 1950s | Show results with:1950s
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Inhomogeneous Electron Gas | Phys. Rev.
It is proved that there exists a universal functional of the density, F ⁡ [ n ⁡ ( r ) ] , independent of v ⁡ ( r ) , such that the expression E ≡ ∫ v ⁡ ( r ) ⁢ ...
[20]
Theoretical studies of enzymic reactions: Dielectric, electrostatic and ...
In this paper we study the stability of the carbonium ion intermediate formed in the cleavage of a glycosidic bond by lysozyme.
[21]
Coarse Grained Model for Semiquantitative Lipid Simulations
This paper describes the parametrization of a new coarse grained (CG) model for lipid and surfactant systems.
[22]
Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal
MMS is an interdisciplinary journal focusing on the fundamental modeling and computational principles underlying various multiscale methods.
[23]
DOE Explains...Exascale Computing - Department of Energy
DOE is deploying the United States' first exascale computers: Frontier at ORNL and Aurora at Argonne National Laboratory and El Capitan at Lawrence Livermore ...Missing: 2023 | Show results with:2023
[24]
Data-driven Solutions of Nonlinear Partial Differential Equations
Nov 28, 2017 · We introduce physics informed neural networks -- neural networks that are trained to solve supervised learning tasks while respecting any given ...
[25]
A Review of Mesh Adaptation Technology Applied to Computational ...
Mesh adaptation techniques can significantly impact Computational Fluid Dynamics by improving solution accuracy and reducing computational costs.
[26]
A Review of Multiscale Computational Methods in Polymeric Materials
Four major blocks are presented in this part: Sequential Multiscale Approaches, Concurrent Multiscale Approaches, Adaptive Resolution Simulations, and Extending ...
[27]
QM/MM through the 1990s: The First Twenty Years of Method ... - NIH
In this article, we trace back the early developments and applications of combined QM/MM techniques for studying complex chemical and enzymatic reactions.Missing: original | Show results with:original
[28]
A mesoscopic bridging scale method for fluids and coupling ... - NIH
A multiscale procedure to couple a mesoscale discrete particle model and a macroscale continuum model of incompressible fluid flow is proposed in this study ...
[29]
Enabling QM-accurate simulation of dislocation motion in 𝛾
Apr 16, 2019 · These results indicate that a QM/MM implementation is a viable tool to investigate dislocation motion in metallic systems, making it possible to ...
[30]
Atomistic simulations of dislocation mobility in refractory high ...
Aug 11, 2021 · We investigate mechanisms underlying the mobilities of screw and edge dislocations in the body-centered cubic MoNbTaW RHEA over a wide temperature range.
[31]
Phase-field modeling of microstructure evolution - ScienceDirect.com
The phase-field (PF) method has emerged as an outstanding tool for simulating the formation and evolution of microstructures during processing.
[32]
[PDF] PHASE-FIELD MODELS FOR MICROSTRUCTURE EVOLUTION
This paper briefly reviews the recent advances in developing phase-field models for various materials processes in- cluding solidification, solid-state ...
[33]
Multiscale crystal-plasticity phase field and extended finite element ...
Mar 1, 2019 · This paper presents a physics-based prediction of crack initiation at the microstructure level using the phase field (PF) model without ...
[34]
Fatigue Damage Prediction in Metallic Materials Based on ...
A multiscale model was used to address the problem of predicting fatigue damage accumulation in metallic materials. Single crystal plasticity served as the ...
[35]
Prediction of multiscale crack propagation in anisotropic ...
In the present work, a more efficient hierarchical cohesive/bulk homogenization approach is proposed for the prediction of multiscale crack propagation in ...
[36]
Multiscale Homogenization Techniques for Predicting Crack ...
Jul 3, 2025 · This paper presents a comprehensive exploration of multiscale homogenization techniques as a powerful toolset for modeling and predicting crack ...<|control11|><|separator|>
[37]
Atomistic QM/MM simulations of the strength of covalent interfaces in ...
May 5, 2020 · Our study focuses on the strength and failure modes of covalently-bonded nanotube–polymer interfaces based on amine, carbene and carboxyl functional groups.
[38]
Multiscale simulations of critical interfacial failure in carbon ...
Dec 11, 2018 · Here, we demonstrate a QM/MM hybrid simulation technique with an automated, dynamic selection of QM regions based on atom-resolved potential ...
[39]
Multiscale modeling of polymer rheology | Phys. Rev. E
Sep 18, 2006 · The stresses calculated from the MD simulations are used in the coarse grained methods, while the coarse grained methods yield velocity ...
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Recent developments on multiscale simulations for rheology and ...
Oct 24, 2024 · Müller and coworkers proposed a three-hierarchical bottom-up multiscale approach starting from the united atom MD (UAMD) model [3]. Their ...
[41]
A surrogate multiscale model for the design of high entropy alloys
Feb 5, 2025 · We propose a temperature and strain-rate dependent dislocation theory model in which the velocity of dislocations is controlled by the average ...
[42]
Frontiers in atomistic simulations of high entropy alloys
Oct 20, 2020 · The field of atomistic simulations of multicomponent materials and high entropy alloys is progressing rapidly, with challenging problems stimulating new ...<|control11|><|separator|>
[43]
The Use of Multiscale Molecular Simulations in Understanding a ...
The present paper focuses on the use of modern molecular dynamics and multi-scale methods of computational biochemistry—from classical molecular dynamics ...
[44]
Reaction-Diffusion Systems in Intracellular Transport & Control
Many signaling pathways in bacteria are two-component systems based on the phosphorylation of two key effector proteins. The primary protein involved in signal ...Missing: multiscale | Show results with:multiscale
[45]
A hybrid model of tumor growth and angiogenesis: In silico ... - NIH
Apr 10, 2020 · We present a hybrid mathematical approach that characterizes vascular changes via an agent-based model, while treating nutrient and VEGF changes through a ...
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A multiscale modeling study of particle size effects on the tissue ...
Nov 25, 2017 · Using a multiscale model, this work investigates particle size effects on the tissue distribution and penetration efficacy of drug-delivery ...Missing: paper | Show results with:paper
[47]
Multiscale mathematical model-informed reinforcement learning ...
Aug 8, 2025 · Mathematical models can aid in quantitatively delineating the evolutionary dynamics of cancer drug resistance and in crafting more efficacious ...Missing: pharmacodynamics | Show results with:pharmacodynamics
[48]
Mechanistic Modeling and Multiscale Applications for Precision ...
May 7, 2020 · Multiscale modeling approaches, which use different layers of biological complexity combining modeling approaches, are exemplified and analyzed ...Bayesian Modeling · Stochastic Modeling · Multiscale Modeling And...Missing: pharmacodynamics | Show results with:pharmacodynamics
[49]
Challenges and Opportunities for Machine Learning in Multiscale ...
The use of ML for multiscale computational modeling presents several challenges that require attention. ... Curse of Dimensionality. ,”. Comput. Meth. Appl. Mech.3.1 Data Dependency · 3.1. 2 Meta-Learning... · 3.2 Discretization Matters...
[50]
Multiscale modeling meets machine learning: What can we learn?
Multiscale modeling can integrate machine learning to create surrogate models, identify system dynamics and parameters, analyze sensitivities, and quantify ...Missing: seminal papers
[51]
A parallel implementation of a mixed multiscale domain ...
This article introduces a mixed domain decomposition method (DDM) designed to meet the requirements of advanced numerical optimization in electrical machines.
[52]
Parallel multi-scale computation using the message passing interface
A master-slave divide-and-conquer approach, emphasizing the functionality and robustness of the code, is implemented through loop parallelism and has reduced ...
[53]
[PDF] Data Management Challenges of Exascale Scientific Simulations
At the exascale, the sheer size of the data can be an issue as many. Petabytes of data can be generated over the course of a few hours. ADIOS. ADIOS is an I/O ...Missing: multiscale | Show results with:multiscale
[54]
Multiscale biomolecular simulations in the exascale era
We discuss the current status and future prospects of multiscale biomolecular simulations on exascale supercomputers with a focus on QM/MM MD.
[55]
[PDF] Virtual Model Validation of Complex Multiscale Systems
Abstract. We propose a virtual statistical validation process as an aid to the design of experiments for the validation of phenomenological models of the ...
[56]
Validation and Trustworthiness of Multiscale Models of Cardiac ...
Feb 14, 2018 · The aim of this article is to clarify potential rationale for the trustworthiness of CEP models by reviewing evidence that has been (or could be) presented to ...
[57]
Uncertainty quantification patterns for multiscale models - Journals
Mar 29, 2021 · In this paper, we will mainly discuss the forward uncertainty propagation which is applicable to both UQ analysis and corresponding SA. The UQ ...
[58]
[PDF] Uncertainty Quantification in Multiscale Materials Modeling
Commonly used nonintrusive UQ methods include Monte Carlo (MC) simulation, global sensitivity analysis (GSA), surrogate models, polynomial chaos, and stochastic.<|separator|>
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A posteriori error estimates for a multi-scale finite-element method
Apr 25, 2021 · In this work, we develop a posteriori error estimates for finite-element multi-scale method for a diffusion problem with a mesh adaptation.
[60]
Variational multiscale a-posteriori error estimation for multi ...
This paper presents an explicit a-posteriori error estimator for the multi-dimensional transport equation based on an approximation to the variational ...
[61]
Multiscale modelling: approaches and challenges - Journals
Aug 6, 2014 · Multiscale systems that are characterized by a great range of spatial–temporal scales arise widely in many scientific domains.
[62]
Seven challenges in the multiscale modeling of multicellular tissues
May 4, 2021 · A common issue faced in multiscale modeling is whether parameters are uniquely identifiable from experimental data in the first place, and the ...Missing: loss | Show results with:loss
[63]
Reproducibility in Computational Neuroscience Models and ...
Reproducibility and replicability are external measures that reflect a user's view of similarities in dynamics between an original reference and a reproduction.
[64]
Credibility assessment of computational models according to ASME ...
This work provides the practical application of the ASME V&V40–2018 risk-based credibility assessment framework, which could be applied to demonstrate model ...
[65]
VV 40 - Assessing Credibility of Computational Modeling through ...
US$85.00 In stockThis Standard provides a framework for assessing the relevance and adequacy of completed V&V activities that establish credibility of a computational model.Missing: multiscale | Show results with:multiscale
[66]
Recent Advances in Machine Learning‐Assisted Multiscale Design ...
Dec 10, 2024 · This review highlights recent advances in machine learning (ML)-assisted multiscale design of energy materials, currently achievable through ...
[67]
Multiscale computational framework linking alloy composition to ...
Jul 15, 2025 · This study introduces a multiscale Integrated Computational Materials Engineering (ICME) framework that combines CALPHAD-based thermodynamic modeling, machine ...
[68]
Multiscale computational modeling techniques in study and design ...
Sep 9, 2024 · This article provides an overview of recent advances, challenges, and opportunities in multiscale computational modeling techniques for study and design of two ...<|control11|><|separator|>
[69]
A Vision for the Future of Multiscale Modeling - ACS Publications
In this paper, we briefly review the NWChem computational chem. suite, including its history, design principles, parallel tools, current capabilities ...