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Interfacing Finite Elements with Deep Neural Operators for Fast Multiscale Modeling of Mechanics Problems

Minglang Yin, Enrui Zhang, Yue Yu, George Em Karniadakis

arXiv:2203.00003v1cs.CEcs.LG

TL;DR

Fine-scale microscopic models capture detailed dynamics but can make multiscale simulation prohibitively expensive. The paper couples a DeepONet surrogate with standard FEM and reports efficient predictions across benchmark mechanics problems, while noting that surrogate errors can cause coupled results to drift or diverge.

  • Problem

    Microscopic models resolve fine-scale behavior but require small spatial and temporal scales, making meso- or macroscale simulations prohibitively expensive.

    Source: §1 · p. 2

  • Method

    The framework couples DeepONet as a surrogate for microscopic behavior with standard FEM for macroscopic response, using concurrent interface coupling.

    Source: §1 · pp. 3–4

  • Results

    DeepONet predicts expensive microscopic behavior in a fraction of a second, and the authors say the framework's overall cost could be shortened by orders of magnitude.

    Source: §4 · p. 24

  • Takeaways & Limitations

    A data-trained neural operator surrogate is promising for learning complex-material dynamics without explicit constitutive models.

    Source: §4 · p. 24

    Surrogate prediction errors may make coupled results drift from the true solution or diverge, particularly amid large microscopic-to-continuum scale differences.

    Source: §4 · p. 25

Abstract

from arXiv · show

Multiscale modeling is an effective approach for investigating multiphysics systems with largely disparate size features, where models with different resolutions or heterogeneous descriptions are coupled together for predicting the system's response. The solver with lower fidelity (coarse) is responsible for simulating domains with homogeneous features, whereas the expensive high-fidelity (fine) model describes microscopic features with refined discretization, often making the overall cost prohibitively high, especially for time-dependent problems. In this work, we explore the idea of multiscale modeling with machine learning and employ DeepONet, a neural operator, as an efficient surrogate of the expensive solver. DeepONet is trained offline using data acquired from the fine solver for learning the underlying and possibly unknown fine-scale dynamics. It is then coupled with standard PDE solvers for predicting the multiscale systems with new boundary/initial conditions in the coupling stage. The proposed framework significantly reduces the computational cost of multiscale simulations since the DeepONet inference cost is negligible, facilitating readily the incorporation of a plurality of interface conditions and coupling schemes. We present various benchmarks to assess accuracy and speedup, and in particular we develop a coupling algorithm for a time-dependent problem, and we also demonstrate coupling of a continuum model (finite element methods, FEM) with a neural operator representation of a particle system (Smoothed Particle Hydrodynamics, SPH) for a uniaxial tension problem with hyperelastic material. What makes this approach unique is that a well-trained over-parametrized DeepONet can generalize well and make predictions at a negligible cost.

1. Introduction

Fine-scale models capture microscopic behavior but can be prohibitively expensive, while existing multiscale approaches use domain decomposition or surrogate models. This work proposes coupling a DeepONet microscopic surrogate with FEM for macroscopic response.

  • Fine-scale models resolve small spatial and temporal scales but make simulating meso- or macroscale material dynamics prohibitively expensive.Source: §1 · p. 2
  • Domain decomposition combines continuum models in the bulk with microscopic models in localized regions, while homogenization uses effective properties whose accuracy and feasibility can be challenging to establish.Source: §1 · pp. 2–3
  • Neural operators can solve PDE systems with new interface conditions or model parameters without retraining, motivating their use as multiscale surrogates, particularly for time-dependent problems.Source: §1 · p. 3
  • The framework couples a DeepONet surrogate of microscopic systems with standard FEM representing macroscopic response, using concurrent coupling.Source: §1 · pp. 3–4

2. Methodology

DeepONet approximates an operator mapping an input function to an output function, while the coupling procedure exchanges interface information between FEM and a fine-scale surrogate. The method permits overlapping or non-overlapping domain decomposition and iterates interface updates until convergence.

2.1. Deep Operator Network (DeepONet)

  • DeepONet's branch network processes a discretized input function and its trunk network processes evaluation coordinates, together approximating an operator from input to output functions.Source: §2.1 · p. 5

2.2. Coupling Methods

  • The coupling divides the domain into macroscopic FEM and microscopic DeepONet regions, and can generalize to other mesh-based or particle models.Source: §2.2 · p. 5
  • Overlapping regions exchange interface conditions on separate boundaries, while non-overlapping regions share one interface for information exchange.Source: §2.2 · p. 6
  • With Robin conditions, the models alternately solve and transmit interface data, updating the boundary condition with relaxation until the stopping criterion is met.Source: §2.2 · p. 6

3. Results

The results evaluate FEM–DeepONet coupling across Poisson, heat-equation, elastoplastic, and hyperelastic problems, including a DeepONet surrogate trained on SPH data. They report prediction accuracy and coupling behavior, including heat-equation stability beyond the training interval, simulation-time reduction for the hyperelastic example, and convergence dependence on the Robin coefficient.

3.1. Poisson equation

  • Poisson coupling errors stay below 1%, with interface relative errors around 0.07% and accurate agreement with the FEM ground truth.The reported maximum absolute error is less than 1%; errors near the interfaces are far less than 1%.Source: §3.1 · pp. 9–11
  • With Dirichlet boundary conditions, displacement errors meet the stopping criterion (L2 error < 2 × 10−3, equivalently relative error < 1%) at iterations 37, 18, and 12 for θ = 0.25, 0.5, and 0.75, respectively.Dirichlet-Aitken converges a bit faster than fixed θ = 0.75. With training on α = 5, relative errors drop from around 10% to less than 1% as test α rises from 1 to 5, then are statistically stable for α > 5; median relative errors decrease slightly from 100 to 300 training cases and then stay statistically stable.Source: §3.1 · pp. 11–12

3.2. Heat equation

  • Heat-equation coupling agrees with ground truth beyond the training interval (t > 1) and remains stable and accurate over long-time integration, although errors grow at a relatively low rate.The framework couples non-overlapping FEM and DeepONet subdomains through interfacial flux in a nested time-step and iteration scheme.Source: §3.2 · pp. 13–14

3.3. Elastoplasticity

  • The elastoplasticity benchmark begins with plane-strain vertical tension of a square solid containing a circular void, using small-deformation, rate-independent plasticity with isotropic hardening.The setup specifies l = 1, ri = 0.1, initial strength Y0 = 0.1, and hardening modulus H0 = 0.3.Source: §3.3 · pp. 15–16
  • In elastoplasticity, DeepONets surrogate the three stress components in an annular void region, while FEM models the surrounding linear-elastic region and exchanges displacement and traction at their interface.FEM interface displacement feeds the DeepONets, whose predicted stress yields traction for the next FEM iteration; traction relaxation uses θ = 0.5.Source: §3.3 · pp. 16–17
  • Below 10% maximum relative error for bulk normal stress and equivalent plastic strain, and below 2% at the interface, with error plateauing by iteration four.Larger bulk stress errors occur at the top edge and bottom corners; the authors attribute these to FEM interpolation or reduced solution regularity.Source: §3.3 · p. 17, Figure 10 · p. 18, §3.4 · p. 18

3.4. Hyperelasticity

  • For hyperelasticity, the framework couples an FEM continuum region with a DeepONet surrogate of an SPH particle region for a vertically fiber-reinforced material under uniaxial tension.The SPH model supplies ground truth for a unit-square plate with a centered void; the material uses the HGO model.Source: §3.4 · pp. 19–20
  • Two DeepONet approaches predict PK1 stress from interface displacement or displacement from interface traction, enabling different interface conditions and coupling methods.The first approach is used in N-D coupling, while both predictions are combined in R-D/R-N coupling.Source: §3.4 · p. 21
  • Approximately 4 hours for an SPH simulation contrasts with a fraction of a second for its DeepONet surrogate in the hyperelastic example.The authors note that this comparison is exacerbated by using a time-dependent SPH solver for a static problem.Source: §3.4 · pp. 21–22, Table 3 · p. 22
  • R = 0.25 converges fastest, whereas the R-N method with R = 5.0 fails to converge, showing that coupling convergence depends on the Robin coefficient.The problem’s nonlinearity generally requires more iterations to reach the stopping criterion.Source: §3.4 · pp. 23–24, Figure 13 · p. 23

4. Discussion

The discussion presents a concurrent FEM–DeepONet coupling framework whose predictions agree well with numerical solutions across four benchmarks. It highlights efficiency benefits and identifies training, coupling, and multiscale-scope challenges that qualify its use.

4. Discussion

  • Across four static and dynamic mechanics benchmarks, coupled predictions agree well with numerical solutions, and Robin conditions tend to converge faster than Neumann or Dirichlet conditions.The framework exchanges interface information between a standard coarse-scale solver and DeepONet until convergence.Source: §4 · p. 24
  • DeepONet inference takes a fraction of a second, and the framework can predict interface information alone when macroscopic dynamics are the main interest.The discussion says overall computational cost could be reduced by orders of magnitude relative to existing multiscale coupling methods.Source: §4 · p. 24
  • Convergence depends on accurate DeepONet training, normalized data, and suitable coupling strategies such as the Robin coefficient.A poorly trained surrogate may slow convergence or cause divergence; a large system condition number may also slow or prevent convergence.Source: §4 · p. 24
  • Large microscopic-to-continuum length and time-scale gaps challenge training and long-term prediction, and inaccurate surrogate predictions may cause drift or divergence.The discussion also identifies noisy training data as a future direction.Source: §4 · p. 25
  • Future extensions include handling geometric variation and fracture progression, and using other coupling methods such as quasicontinuum, multigrid, or HMM.For fracture, the field of interest may move with the crack tip, challenging both coupling and network training.Source: §4 · p. 25

Appendix A. Smoothed Particle Hydrodynamics

The appendix introduces Total Lagrangian Smoothed Particle Hydrodynamics (TLSPH), approximating physical quantities and their gradients from neighboring particles through a kernel. It describes correction procedures and the solid-equilibrium formulation used in the SPH model.

Appendix A. Smoothed Particle Hydrodynamics

  • TLSPH approximates physical quantities from neighboring particles using a kernel, with integration approximated numerically using particle volumes and a third-order polynomial weighting kernel.The kernel radius is denoted by h.Source: Appendix A · p. 25
  • The appendix introduces gradient corrections to preserve symmetrization and first-order completeness, plus a correction to suppress spurious hourglassing.A shape tensor is used in the corrected gradient formulation.Source: Appendix A · p. 26
  • The SPH formulation applies the solid equilibrium equation using the first Piola-Kirchhoff stress, body force, referential density, and acceleration.Internal forces are described as emerging from the divergence of the first Piola-Kirchhoff stress.Source: Appendix A · p. 26
  • The appendix also specifies a fiber-enhanced tissue strain-energy function and defines its principal invariants, while approximating the deformation derivative in time.The material parameters are set to match the FEM model in Section 3.4.Source: Appendix A · p. 27

Appendix B.1. Accuracy of SPH

In the SPH–FEM benchmark, interface displacements and tractions generally correspond well, with deviations near two angles attributed partly to differences in numerical integration.

Appendix B.1. Accuracy of SPH

  • FEM and SPH interface displacements and tractions correspond well overall, with deviations around θ = π/2 and θ = 3π/2.The comparison uses the same benchmark problem and a distributed traction on the top edge; the figure plots interfacial displacement and traction against angle.Source: Appendix B.1 · p. 27
  • The benchmark applies a distributed traction at the top edge and a Dirichlet-type displacement condition at the FEM–SPH interface.The plotted FEM tractions are solid lines and SPH tractions are dashed lines.Source: Appendix B.1 · p. 27
  • Differences in numerical integration partially contribute to the overall errors observed in Section 3.4.Source: Appendix B.1 · p. 27

Appendix C. Network Training Details

The training details specify a shared fully connected architecture, problem-dependent branch input sizes and training durations, and comparatively small training–testing error differences in reported cases.

Appendix C. Network Training Details

  • All subnetworks use four-layer fully connected networks with three hidden layers, tanh activation, and 100 or 150 neurons per hidden layer.Branch input dimensions range from 51 to 120 interface points.Source: Appendix C · p. 28
  • Training runs for 100,000 epochs for the heat and Poisson examples and 1,000,000 epochs for the other two examples.Source: Appendix C · p. 28
  • Reported cases show relatively small differences between training and testing errors, which the authors interpret as indicating good network generalization.The passage also refers to case-specific training and testing MSEs.Source: Appendix C · p. 28

Appendix D.1. Random field generation

The appendix outlines FFT-based random-field generation from Gaussian white noise. The method uses a correlation function, with α = 5 for training data in Sections 3.1, 3.2, and 3.4.

Appendix D.1. Random field generation

  • Random fields φ(x) are sampled from Gaussian white noise W(x) using the FFT-based generation algorithm.The passage introduces the generation procedure but does not include its displayed formula.Source: Appendix D.1 · p. 28
  • The correlation function γ is ∥k∥−α, where ∥k∥ is the L2 norm of wave number k.Source: Appendix D.1 · p. 28
  • Training data in Sections 3.1, 3.2, and 3.4 use this method with α = 5.The passage directs readers to [40] for further theoretical details.Source: Appendix D.1 · p. 28

Appendix D.2. Sampling in elastoplasticity

For elastoplasticity, non-uniform tension is generated from independent normal random variables. The standard deviation is set to balance dataset cases with and without plastic deformation.

Appendix D.2. Sampling in elastoplasticity

  • Figure C.16 presents training/testing error histories for different problems, while Figure C.17 gives network inputs, outputs, and MSE errors for each example.Source: Figure C.16 · p. 29, Figure C.17 · p. 30
  • The tension-field sampling uses independent A_i and B_i drawn from N(0, 0.05^2), with standard deviation 0.05.Source: Appendix D.2 · pp. 28, 30
  • The standard deviation is chosen to produce similar numbers of dataset cases with and without plastic deformation.Source: Appendix D.2 · p. 30

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