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A physics-informed deep learning approach for solving strongly degenerate parabolic problems

Pasquale Ambrosio, Salvatore Cuomo, Mariapia De Rosa

TL;DR

This work demonstrates that Physics-Informed Neural Networks can accurately approximate solutions to strongly degenerate parabolic PDEs with asymptotic Laplacian-type structure arising in gas filtration. By training a four-layer, 20-neuron FF-DNN to minimize PDE residuals and boundary losses, the authors achieve high-fidelity predictions across two- and three-dimensional domains, including long-time behavior, for five test problems with known exact solutions. They analyze both direct PINN accuracy and behavior under a regularized approximating problem, reporting time-dependent $L^{2}$-errors and relative errors that are typically in the $10^{-6}$–$10^{-4}$ range, even as $T$ grows large. The work highlights PINNs’ flexibility to adapt to varying initial-boundary data and parameter regimes, while acknowledging limitations in non-differentiable regions and contour precision, and it positions PINNs as a competitive alternative to finite-difference methods for complex, degenerate PDEs with known solutions. This has potential practical impact for gas filtration modeling and other applications involving degenerate diffusion with Laplacian-type asymptotics, where fast, adaptable solvers are valuable.

Abstract

In recent years, Scientific Machine Learning (SciML) methods for solving partial differential equations (PDEs) have gained increasing popularity. Within such a paradigm, Physics-Informed Neural Networks (PINNs) are novel deep learning frameworks for solving initial-boundary value problems involving nonlinear PDEs. Recently, PINNs have shown promising results in several application fields. Motivated by applications to gas filtration problems, here we present and evaluate a PINN-based approach to predict solutions to strongly degenerate parabolic problems with asymptotic structure of Laplacian type. To the best of our knowledge, this is one of the first papers demonstrating the efficacy of the PINN framework for solving such kind of problems. In particular, we estimate an appropriate approximation error for some test problems whose analytical solutions are fortunately known. The numerical experiments discussed include two and three-dimensional spatial domains, emphasizing the effectiveness of this approach in predicting accurate solutions.

A physics-informed deep learning approach for solving strongly degenerate parabolic problems

TL;DR

This work demonstrates that Physics-Informed Neural Networks can accurately approximate solutions to strongly degenerate parabolic PDEs with asymptotic Laplacian-type structure arising in gas filtration. By training a four-layer, 20-neuron FF-DNN to minimize PDE residuals and boundary losses, the authors achieve high-fidelity predictions across two- and three-dimensional domains, including long-time behavior, for five test problems with known exact solutions. They analyze both direct PINN accuracy and behavior under a regularized approximating problem, reporting time-dependent -errors and relative errors that are typically in the – range, even as grows large. The work highlights PINNs’ flexibility to adapt to varying initial-boundary data and parameter regimes, while acknowledging limitations in non-differentiable regions and contour precision, and it positions PINNs as a competitive alternative to finite-difference methods for complex, degenerate PDEs with known solutions. This has potential practical impact for gas filtration modeling and other applications involving degenerate diffusion with Laplacian-type asymptotics, where fast, adaptable solvers are valuable.

Abstract

In recent years, Scientific Machine Learning (SciML) methods for solving partial differential equations (PDEs) have gained increasing popularity. Within such a paradigm, Physics-Informed Neural Networks (PINNs) are novel deep learning frameworks for solving initial-boundary value problems involving nonlinear PDEs. Recently, PINNs have shown promising results in several application fields. Motivated by applications to gas filtration problems, here we present and evaluate a PINN-based approach to predict solutions to strongly degenerate parabolic problems with asymptotic structure of Laplacian type. To the best of our knowledge, this is one of the first papers demonstrating the efficacy of the PINN framework for solving such kind of problems. In particular, we estimate an appropriate approximation error for some test problems whose analytical solutions are fortunately known. The numerical experiments discussed include two and three-dimensional spatial domains, emphasizing the effectiveness of this approach in predicting accurate solutions.
Paper Structure (11 sections, 2 theorems, 70 equations, 11 figures, 12 tables)

This paper contains 11 sections, 2 theorems, 70 equations, 11 figures, 12 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Notation and preliminaries
  4. Physics-informed methodology
  5. Numerical results
  6. First test problem
  7. Second test problem
  8. Third test problem
  9. Fourth test problem
  10. Fifth test problem
  11. Conclusions

Key Result

Theorem 2.3

Let $n\geq2$, $\frac{2n\,+\,4}{n\,+\,4}\leq q<\infty$ and $f\in L^{q}\left(0,T;W^{1,q}(\Omega)\right)$. Moreover, assume that is a weak solution of equation $\mathrm{(eq:mainprob)_{1}}$. Then the solution satisfies Furthermore, the following estimate holds true for any parabolic cylinder $Q_{\rho}(z_{0})\subset Q_{R}(z_{0})\subset Q_{R_{0}}(z_{0})\Subset\Omega_{T}$ and a positive constant $c$ d

Figures (11)

  • Figure 3.1: Overall structure of the proposed methodology. An FF-DNN serves as the neural network's architecture. Automatic differentiation is employed to calculate the loss terms associated with the model's dynamics. The loss function comprises two components: the physics loss, represented by $\mathcal{L}_{\mathcal{F}}$, and the boundary loss denoted by $\mathcal{L}_{\mathcal{B}}$. During the optimization phase, the objective is to minimize the loss function with respect to the set of hyperparameters $\theta$.
  • Figure 4.1: Plots of the exact solution to problem (\ref{['eq:P1']}) (above) and the predicted solution $\hat{u}(\cdot,t)$ (below) for $t=0$ (left), $t=2.5$ (center) and $t=4.5$ (right).
  • Figure 4.2: Superposition of the level curves of the exact solution $u(\cdot,t)$ and the predicted solution $\hat{u}(\cdot,t)$ for $t=0$ (left), $t=5$ (center) and $t=10$ (right). The contour lines corresponding to the same level are almost indistinguishable for any fixed $t\in[0,10]$.
  • Figure 4.3: Plots of the approximate solution $\hat{u}_{\varepsilon}(\cdot,t)$ for $\varepsilon=10^{-3}$ (above) and $\varepsilon=10^{-9}$ (below), at times $t=0$ (left), $t=5$ (center) and $t=10$ (right). Here $\Omega'=\Omega$ and $[t_{1},t_{2}]=[0,10]$.
  • Figure 4.4: Superposition of the level curves of the exact solution $u(\cdot,t)$ and the predicted solution $\hat{u}_{\varepsilon}(\cdot,t)$ for $\varepsilon=10^{-3}$ (above) and $\varepsilon=10^{-9}$ (below), at times $t=0$ (left), $t=5$ (center) and $t=10$ (right). Here $\Omega'=\Omega$ and $[t_{1},t_{2}]=[0,10]$. For every $\varepsilon\in[10^{-9},10^{-3}]$, the contour lines of the approximate solution $\hat{u}_{\varepsilon}(\cdot,t)$ perfectly overlap those of $u(\cdot,t)$ for any fixed $t\in[0,10]$.
  • ...and 6 more figures

Theorems & Definitions (6)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Definition 2.5
  • Definition 2.6