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Physics-informed Neural Networks for Solving Nonlinear Diffusivity and Biot's equations

Teeratorn Kadeethum, Thomas M Jorgensen, Hamidreza M Nick

arXiv:2002.08235v1cs.CEcs.LGcs.NAmath.NA

TL;DR

Conventional numerical methods can be computationally demanding and mesh-sensitive, motivating PINNs for forward and inverse nonlinear diffusivity and Biot problems. The paper evaluates accuracy across training-data and hyperparameter choices, reporting improved accuracy with more examples and parameter-estimation feasibility in the Biot inverse case.

  • Problem

    Conventional methods can require substantial computational resources and depend heavily on mesh quality, potentially limiting their suitability for inverse problems or complex geometries.

    Source: Introduction · p. 2

  • Method

    The paper applies physics-informed neural networks to forward and inverse nonlinear diffusivity and Biot equations, with inverse training estimating unknown physical parameters.

    Source: Introduction · p. 2, Inverse model of diffusivity equation · p. 11

  • Results

    Across diffusivity and Biot cases, more training examples improved PINN accuracy; with 1,000 Biot inverse examples, average parameter-estimation error was on the order of 1%, despite large variation.

    Source: Verification of the forward model · pp. 8, 15, Inverse model of Biot’s equations · p. 19

  • Takeaways & Limitations

    The authors report that PINNs can estimate unknown Biot physical parameters with a reasonably sized dataset, while trained models show substantial variation.

    Source: Inverse model of Biot’s equations · p. 19

    The study leaves incomplete inputs untested, including cases where displacement and pressure are unavailable at the same spatial and temporal coordinates.

    Source: Conclusion · p. 20

Abstract

from arXiv · show

This paper presents the potential of applying physics-informed neural networks for solving nonlinear multiphysics problems, which are essential to many fields such as biomedical engineering, earthquake prediction, and underground energy harvesting. Specifically, we investigate how to extend the methodology of physics-informed neural networks to solve both the forward and inverse problems in relation to the nonlinear diffusivity and Biot's equations. We explore the accuracy of the physics-informed neural networks with different training example sizes and choices of hyperparameters. The impacts of the stochastic variations between various training realizations are also investigated. In the inverse case, we also study the effects of noisy measurements. Furthermore, we address the challenge of selecting the hyperparameters of the inverse model and illustrate how this challenge is linked to the hyperparameters selection performed for the forward one.

Introduction

The paper motivates PINNs for nonlinear multiphysics problems where conventional numerical methods can be costly and mesh-sensitive, and proposes using them for forward and inverse nonlinear diffusivity and Biot equations.

Introduction

  • Conventional numerical methods can require substantial computational resources and depend heavily on mesh quality, potentially limiting their suitability for inverse problems or complex geometries.Source: Introduction · p. 2
  • PINNs encode physical information as regularization in neural-network loss functions, which prior work reports can support good accuracy with reasonably sized datasets.Source: Introduction · p. 2
  • The paper applies PINNs to forward and inverse problems for nonlinear diffusivity and Biot’s equations.The authors also investigate training-example size, hyperparameters, and stochastic variation across training realizations.Source: Introduction · p. 2
  • The coupled fluid-flow and solid-deformation process is highly nonlinear and often involves complex geometries, motivating its study with PINNs.Source: Introduction · p. 2

Methodology

The methodology defines nonlinear Biot and diffusivity settings, then trains neural networks with data-fit and physics-informed loss terms. Forward and inverse models differ in their available training data and whether physical parameters are known.

Governing equations

  • The Biot setup distinguishes displacement and traction boundaries for solid deformation from pressure and flux boundaries for fluid flow, with Dirichlet and Neumann boundary types identified.Source: Governing equations · pp. 2–3
  • The conductivity tensor maps pressure-gradient components to fluid-velocity components; its off-diagonal terms are set to zero under an isotropic, homogeneous porous-medium assumption.The diffusivity equation is a special case of Biot’s equations when α = 0.Source: Governing equations · p. 3

PINN architecture

  • The network’s input and output sizes follow the problem formulation, while hidden-layer count and neuron count are treated as hyperparameters learned alongside weights and biases.The study uses the same number of neurons in each hidden layer and reports comparable ReLU and tanh results, presenting tanh results.Source: Neural network architecture · pp. 3–4, Figure 1 · p. 4

PINN loss

Training data

  • Forward training uses boundary and initial data for the data-fit term and collocation points to evaluate physics-informed terms; inverse training uses measured domain points for both.Inverse models may also use unmeasured domain collocation points for the physics-informed term alone.Source: Loss function definition · p. 5, Figure 2 · p. 6

PINN architecture

  • Automatic differentiation estimates output derivatives at collocation points from the current network, allowing the physics-informed residual to depend on network weights, biases, inputs, and physical parameters.Source: Training the PINN · p. 5

Forward and inverse modeling

  • Forward and inverse models both learn an input-to-output mapping, but inverse training also adjusts unknown physical parameters alongside network weights and biases.The inverse loss does not compare estimated parameters against ground-truth values; parameter convergence is expected through minimizing the combined loss.Source: Training the PINN · pp. 5–6

Hyperparameter selection

  • Forward-model hyperparameters may inform inverse-model choices when training-set sizes are similar, while validation-based selection would reserve some measurements from training.Source: Training the PINN · p. 6

Results and Discussion

The Results and Discussion section examines PINN accuracy in forward problems and parameter recovery in diffusivity and Biot inverse problems. It considers training-set size, noisy measurements, and variation across realizations, with the reported effects differing between state-variable accuracy and physical-parameter estimates.

Verification of the forward model

  • For diffusivity forward modeling, increasing Nb and NΠ improved average accuracy, with their combined count in the hundreds yielding average L2 error below 1.0 × 10^-2.Without Π regularization, 96 training examples gave average relative L2 error 5.6 × 10^-2; including Π reduced average L2 error below 1.0 × 10^-2.Source: Verification of the forward model · pp. 8–9, Table 1 · p. 9
  • For diffusivity forward modeling, Nhl = 6 and Nn = 5 gave the best explored accuracy; larger or smaller architectures showed overfitting or underfitting, and test relative L2 error was 1.43 × 10^-3.This configuration used Nb = 96 and NΠ = 160, and its test error was comparable to validation accuracy.Source: Verification of the forward model · p. 10, Table 2 · p. 10

Inverse model of diffusivity equation

  • For diffusivity inversion, Ntr = 200 was needed to avoid substantial stochastic fluctuations, while Ntr > 200 gave average θ1 and θ2 percentage errors below 1% but with large standard deviations.Across 24 realizations at Ntr = 200, the minimum errors were 2.31 × 10^-2 and 7.33 × 10^-2, and the maximum errors were 1.93 × 10^-1 and 4.92 × 10^-1, for θ1 and θ2, respectively.Source: Inverse model of diffusivity equation · p. 11, Figure 6 · p. 12
  • For diffusivity inversion, error increased with noise and decreased as Ntr increased, so higher noise required more training data to estimate physical parameters accurately.Table 4 reports average percentage errors for θ1 and θ2 across noise levels and training-set sizes, averaged over 10 realizations.Source: Inverse model of diffusivity equation · p. 12, Table 4 · p. 13

Inverse model of Biot’s equations

Verification of the forward model

  • For Biot forward modeling, Nhl = 6 and Nn = 20 yielded average test L2 error 9.05 × 10^-5, comparable to validation error 9.08 × 10^-5.The comparison averages 27 networks trained with Nb = 96 and NΠ = 160.Source: Verification of the forward model · p. 15
  • During Biot forward-model convergence, MSEb was generally higher than MSEΠu and MSEΠp, which were comparable, while the loss terms showed minor oscillations.The behavior was reported for Nhl = 6, Nn = 20, Nb = 96, and NΠ = 160.Source: Verification of the forward model · p. 15

Results and Discussion

  • Using Nhl = 6 and Nn = 20, selected from forward-model sensitivity analysis, yielded the least L2 error for p, u, and v and the lowest percentage error for θ1–θ5; errors were not much different across hyperparameter combinations.Source: Inverse model of Biot’s equations · p. 18
  • Below 0.01%: relative L2 errors of u, v, and p were always below this level; at Ntr = 1000, average physical-parameter estimation error was about 1%, with substantial variation across realizations.Across 27 realizations with Ntr = 1000, percentage errors for θ1–θ5 ranged respectively from 0.05 to 2.51, 0.02 to 6.14, 0.03 to 4.75, 0.04 to 8.12, and 0.03 to 3.60. Stochastic variation in the estimated parameters generally decreased as the training set grew.Source: Inverse model of Biot’s equations · pp. 18–19
  • As measurement noise increased, percentage errors in θ1–θ5 increased, but larger Ntr reduced the noise impact.The results were averaged over ten realizations.Source: Inverse model of Biot’s equations · p. 19

Conclusion

The study concludes that PINNs solve the tested forward and inverse nonlinear diffusivity and Biot’s equation problems, while inverse estimates vary substantially across realizations. Noise tolerance and transfer of forward hyperparameter choices to inverse modeling are supported within the tested cases.

Conclusion

  • 9.05 × 10−5 ± 3.1 × 10−4 was the average L2 error for test-set displacement and pressure predictions across 27 forward-model realizations.The conclusion limits this result to the types of geometries considered in the paper.Source: Conclusion · p. 19
  • In inverse modeling, PINNs predicted all unknown physical parameters with around 1% average percentage error, while estimates across 27 Biot-equation models trained on 1000 examples ranged from 0.02 to 8.12.Increasing the number of training examples reduced this variability, and averaging PINN models with different random initializations was essential.Source: Conclusion · p. 19
  • At noise levels up to 5%, inverse parameter-estimation error was approximately below 15%, with results improving as training examples increased.The conclusion again specifies averaging across several PINN realizations.Source: Conclusion · p. 20
  • Forward-model hyperparameter selection was experimentally applicable to inverse modeling in the cases considered, but requires further study on other nonlinear partial differential equations.For the Biot inverse case, the study used the forward sensitivity analysis choice of 6 hidden layers and 20 neurons per layer.Source: Conclusion · p. 20, Table 7 · p. 20
  • Future work includes testing PINNs when displacement and pressure data are unavailable at the same spatial and temporal coordinates.The authors also propose studying mini-batch training and smarter weight-and-bias initialization based on transfer-learning principles.Source: Conclusion · p. 20
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