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Physics-Informed Neural Networks for Biot's Model via Fixed-Stress Splitting and Energy Natural Gradient Descent

Kexin Sun, Qiang Liu, Minfu Feng, Mingchao Cai

arXiv:2608.26303v1math.NAmath.OC

TL;DR

Biot’s strongly coupled poromechanics creates ill-conditioned PINN optimization and near-incompressible formulations can suffer locking and pressure oscillations. The paper proposes FS-ENGD-PINN, combining fixed-stress splitting, Energy Natural Gradient Descent, and a three-field formulation; benchmarks report improved accuracy, efficiency, and robustness. Its trade-off is higher per-epoch computational cost and memory from constructing and approximating the Energy Gram matrix.

  • Problem

    Strong coupling, multiscale behavior, saddle-point structure, and ill-conditioning make Biot’s model difficult for PINNs, while standard formulations face locking and pressure oscillations near incompressibility.

  • Method

    FS-ENGD-PINN combines fixed-stress splitting for physical decoupling, Energy Natural Gradient Descent for optimization, and a three-field mixed formulation with independent total pressure.

  • Results

    Numerical benchmarks show FS-ENGD-PINN outperforms baseline methods in time-to-accuracy and final solution quality while resolving challenging singularities, interfaces, and incompressible-limit behavior.

  • Takeaways & Limitations

    The framework provides a decoupled learning-based solver for Biot’s poroelasticity that maintains stability and robustness across heterogeneous media, three-dimensional settings, and nearly incompressible regimes.

  • Takeaways & Limitations

    ENGD increases computational cost and memory per epoch because constructing and approximating the Gram matrix is required.

Abstract

from arXiv · show

Physics-Informed Neural Networks (PINNs) have recently gained considerable attention as a mesh-free framework for solving partial differential equations. Nevertheless, their performance deteriorates when applied to strongly coupled multiphysics systems, such as Biot's consolidation model, due to severely ill-conditioned optimization landscapes. In this work, we propose a robust PINN-based solver, termed FS-ENGD-PINN, which synergistically integrates physics-based decoupling with geometry-aware optimization. Specifically, the Fixed-Stress (FS) splitting scheme is employed to decompose the coupled poroelastic system into contractive mechanics and flow subproblems, thereby significantly improving training stability and convergence. To further accelerate optimization, we adopt Energy Natural Gradient Descent (ENGD), which approximates the Newton direction in function space effectively mitigates stiffness-induced slow convergence. Moreover, to address volumetric locking arising in the nearly incompressible regime, we incorporate a three-field mixed formulation with an additional total pressure variable into the PINN framework. Extensive numerical experiments demonstrate that the proposed FS-ENGD-PINN consistently outperforms standard PINN formulations in terms of accuracy and robustness, providing a unified and reliable learning-based solver for poroelasticity across a wide range of material parameters.

1 Introduction

Biot’s strongly coupled poromechanics challenges both conventional numerical methods and PINN optimization. FS-ENGD-PINN combines fixed-stress decoupling, energy natural-gradient optimization, and a mixed formulation, with benchmarks reporting improved efficiency, quality, and robustness.

  • Biot’s model couples fluid flow and solid deformation, making its governing PDE system difficult to solve because of strong interactions and complex physical mechanisms.
  • Standard formulations can exhibit pressure oscillations and volumetric locking near the incompressible limit.
  • Strongly coupled Biot-type systems create ill-conditioned PINN learning problems through multiscale behavior, saddle-point structure, and tightly coupled fields.
  • FS-ENGD-PINN combines fixed-stress splitting with Energy Natural Gradient Descent to address stiffness from physical coupling and numerical optimization.
  • The framework links fixed-stress contraction at the physical level with natural-energy-gradient optimization efficiency.
  • The three-field mixed PINN treats total pressure as an independent output to address volumetric locking and pressure oscillations near ν →0.5.
  • Benchmarks report that FS-ENGD-PINN outperforms baseline methods in time-to-accuracy and final solution quality while remaining robust for heterogeneous media with sharp interfaces.

2 Mathematical Formulations

The mathematical formulation first describes coupled displacement–pressure Biot equations and their boundary and initial data, then introduces a three-field mixed formulation for nearly incompressible materials. The formulation uses total pressure as an internal variable and recovers the incompressible limit without numerical blow-up.

  • 2.1 Two-field Formulation of Biot’s Model: The quasi-static Biot model couples fluid flow and solid deformation through PDEs posed on a bounded domain in two or three spatial dimensions.
  • 2.1 Two-field Formulation of Biot’s Model: The two-field formulation uses solid displacement u and fluid pressure p as its primary unknowns.
  • 2.1 Two-field Formulation of Biot’s Model: The total stress is σtotal = σ(u) − αpI, with coefficients including storage c0, Biot–Willis coefficient α, and hydraulic conductivity κ.
  • 2.1 Two-field Formulation of Biot’s Model: The formulation includes initial conditions and Dirichlet or Neumann boundary conditions for displacement and pressure, with sufficiently smooth data and homogeneous conditions in the analysis.
  • 2.2 Three-field Formulation for Incompressible Limit: Near ν→0.5 or λ→∞, standard formulations and basic PINNs can suffer volumetric locking, pressure oscillations, instability, and solution blow-up.
  • 2.2 Three-field Formulation for Incompressible Limit: The three-field formulation introduces total pressure ξ as an independent unknown, eliminating the singular λ∇·u term from the momentum equation.
  • 2.2 Three-field Formulation for Incompressible Limit: Its constraint weakly enforces the relation among volumetric strain, pore pressure, and total pressure.
  • 2.2 Three-field Formulation for Incompressible Limit: As λ→∞, the mixed system recovers ∇·u→0 without numerical instability or solution blow-up, while ξ requires no independent boundary condition and lies in L2(Ω).

3 Methodology: The FS-ENGD-PINN Framework

FS-ENGD-PINN combines fixed-stress splitting for contractive physical subproblems with Energy Natural Gradient Descent for geometry-aware optimization. Its analysis establishes unconditional stability for the three-field split and asymptotic convergence of the network approximations.

  • Fixed-Stress Splitting: Fixed-stress splitting decomposes Biot’s coupled system into sequential pressure and mechanics subproblems.The three-field formulation alternates between a pressure subproblem and a coupled displacement–total-pressure subproblem, with intrinsic stabilization.
  • Fixed-Stress Splitting: The three-field fixed-stress iteration converges to the unique solution for any positive Lamé parameters.Its unconditional convergence follows from the contraction property under the specified stabilization relation.
  • PINN Formulation: The PINN maps trainable parameters to functions and minimizes an energy-based loss enforcing PDE, boundary, and initial conditions.The framework uses fully connected networks with differentiable activations and constructs losses from the relevant collocation-point residuals.
  • Energy Natural Gradient Descent: Energy Natural Gradient Descent uses the energy Hessian geometry to approximate the Newton direction in function space.For quadratic energies with a unique minimizer, its update direction corresponds to the projected true error on the tangent space.
  • Convergence Analysis: ENGD attains a tighter effective error bound than Adam but incurs additional Gram-matrix computation and memory cost per epoch.The reported trade-off is fewer epochs for high-fidelity convergence versus greater per-epoch overhead.
  • Convergence Analysis: FS-ENGD-PINN’s network approximations converge asymptotically in time-derivative norms, including total pressure in the three-field case.The error is bounded using the fixed-stress contraction factor and coupling constants.
  • Convergence Analysis: Dynamic resampling can produce early oscillations and late micro-oscillations near sharp heterogeneous interfaces, although the paper reports physical validation for this setting.Homogeneous-media training is described as predominantly monotonic with simple static weights.

4 Numerical Benchmarks

The numerical-benchmark section evaluates FS-ENGD-PINN’s accuracy, efficiency, and robustness on Biot-equation test problems.

  • Numerical Validation: Numerical benchmarks are used to validate the accuracy, efficiency, and robustness of FS-ENGD-PINN for solving Biot’s equations.The supplied passage introduces the validation program without reporting individual benchmark results.

4.1 Implementation Details

The implementation uses dynamically resampled collocation points, fully connected tanh networks, efficient ENGD computations, and comparisons with two PINN baselines.

  • Sampling: Collocation points are regenerated from a uniform distribution at the start of each fixed-stress iteration.The strategy applies to PDE residuals, boundary conditions, and initial conditions.
  • Network Configuration: Pressure and displacement networks use fully connected architectures with tanh activations and Xavier initialization.The default configuration has two hidden layers of 32 neurons for each separate network.
  • Gradient Computation: The Gram matrix is computed through Jacobian contraction, and the natural energy-gradient direction is obtained by solving a least-squares problem.These choices avoid explicit per-sample outer products and direct Gram-matrix pseudoinversion.
  • Gradient Computation: Least-squares computation of the update direction improves numerical stability when the Gram matrix is ill-conditioned or singular.
  • Optimization Schedule: The ENGD learning rate is selected dynamically by logarithmic line search within (0,1] at each update step.
  • Optimization Schedule: The default schedule allows 20 fixed-stress iterations with 20 ENGD epochs per iteration.The passage notes that this applies unless otherwise specified.
  • Baselines and Metrics: The framework is compared with Adam-based standard FS-PINN and unsplit ENGD-based coupled PINN using Relative L2 Error and H1 Error.
  • Computational Environment: Experiments are implemented in Python with JAX and executed on an NVIDIA RTX A6000 GPU.

4.2 Validation on Standard Benchmarks (Two-field Model)

FS-ENGD-PINN is validated on standard two-field Biot benchmarks spanning mixed-boundary 2D, 3D, complex spatiotemporal, Mandel, and heterogeneous layered problems. It achieves accurate solutions and a favorable accuracy-efficiency trade-off against FS-PINN and coupled ENGD-PINN.

  • Example 1: 2D Linear Biot’s Problem with Mixed BCs: The 2D benchmark compares predicted and exact pressure and displacement fields at T_final under mixed Dirichlet and Neumann boundary conditions.The framework is evaluated on Ω=[0,1]×[0,1] with default material parameters and T_final=1.0.
  • Comparative Analysis: FS-PINN with Adam reaches 252.62 s runtime but stagnates near relative L2 errors of 10^-4 for pressure and 10^-3 for displacement.The reported accuracy remains insufficient for high-precision requirements despite substantially more training epochs.
  • Comparative Analysis: FS-ENGD-PINN exceeds the coupled ENGD-PINN’s final converged accuracy by the 15th FS iteration, equivalent to 300 epochs and 293.77 s.The full-cycle runtime is 385.71 s for FS-ENGD-PINN versus 317.18 s for coupled ENGD-PINN, but the proposed method reaches the target accuracy sooner.
  • Example 2: 3D Linear Biot’s Problem: The framework achieves excellent agreement with analytical solutions on the 3D benchmark, demonstrating capability on higher-dimensional problems.The 3D domain is Ω=[0,1]^3, with 2000 interior, 1000 Dirichlet-boundary, 500 Neumann-boundary, and 1000 initial-condition collocation points.
  • Example 3: Problem with Complex Spatiotemporal Solution: Across 3D and complex spatiotemporal scenarios, FS-ENGD-PINN consistently achieves low relative L2 and H1 errors.The complex benchmark uses trigonometric solutions and mixed traction-flux boundary conditions.
  • Example 4: The Mandel Problem: The solver captures the Mandel-Cryer pressure evolution, including its characteristic initial rise followed by a subsequent drop.Predicted and exact normalized solutions agree at t=1.0 for the Mandel problem.

4.3 Performance near the Incompressible Limit (Three-field Model)

Near the incompressible limit, the two-field model develops severe volumetric locking and fails to resolve displacement, while the three-field formulation remains accurate and stable.

  • Two-field model: At ν=0.4999999, pressure errors converge rapidly but displacement errors stagnate near 1.0.The convergence history shows this decoupling between accurate pressure and failed mechanical deformation.
  • Three-field model: The three-field formulation maintains exceptional accuracy and stability across the tested Poisson’s-ratio range.It introduces total pressure ξ as an independent variable to alleviate volumetric locking and support accurate near-incompressible poroelastic solutions.
  • Three-field model: At ν=0.4999999, the three-field model achieves relative L2 displacement errors on the order of 10^-4 and H1 errors on the order of 10^-3.These results are reported in the comparison of two-field and three-field models near the incompressible limit.

5 Conclusion and Future Work

The paper concludes that FS-ENGD-PINN combines Fixed-Stress splitting and Energy Natural Gradient Descent to address coupling and optimization stiffness in Biot’s poroelasticity. It reports robust performance across challenging benchmarks, while identifying broader simulations, nonlinear models, inverse problems, and neural-operator generalization as future directions.

  • Conclusion: FS-ENGD-PINN integrates Fixed-Stress splitting with Energy Natural Gradient Descent in a decoupled learning-based framework for Biot’s poroelasticity.The two-level approach targets stiffness from multiphysics coupling and ill-conditioned optimization.
  • Conclusion: Numerical benchmarks resolve sharp initial temporal singularities, the non-monotonic Mandel-Cryer effect, and internal weak discontinuities in layered media.The framework also maintains scalability in three dimensions.
  • Conclusion: The three-field mixed formulation eliminates spurious pressure oscillations and precludes volumetric locking near the incompressible limit.
  • Future Work: Future work includes field-scale heterogeneous reservoirs, fluid-driven fracture propagation, nonlinear poromechanics, inverse problems, and neural operators.The listed directions also cover deformation-dependent permeability, sparse or noisy observations, and varying parameter ranges and geometries.
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