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Approximately well-balanced Discontinuous Galerkin methods using bases enriched with Physics-Informed Neural Networks

Emmanuel Franck, Victor Michel-Dansac, Laurent Navoret

TL;DR

This work introduces a DG method enhanced by basis enrichment with priors obtained from parametric PINNs to achieve approximate well-balancedness for hyperbolic balance laws. By replacing or multiplying basis functions with a learned steady-state prior, the scheme preserves the convergence order while reducing the error constant near steady states, as demonstrated across linear and nonlinear test problems in one and two dimensions. Theoretical results confirm convergence for non-polynomial bases and quantify prior-dependent error bounds, while numerical experiments across linear advection, shallow water, and Euler–Poisson systems show substantial gains in steady-state accuracy and robustness to perturbations. The offline PINN prior construction coupled with online DG basis enrichment yields improved accuracy with modest online costs, suggesting broad applicability and avenues for future extensions such as orthogonalization, neural operators, and time-dependent priors.

Abstract

This work concerns the enrichment of Discontinuous Galerkin (DG) bases, so that the resulting scheme provides a much better approximation of steady solutions to hyperbolic systems of balance laws. The basis enrichment leverages a prior - an approximation of the steady solution - which we propose to compute using a Physics-Informed Neural Network (PINN). To that end, after presenting the classical DG scheme, we show how to enrich its basis with a prior. Convergence results and error estimates follow, in which we prove that the basis with prior does not change the order of convergence, and that the error constant is improved. To construct the prior, we elect to use parametric PINNs, which we introduce, as well as the algorithms to construct a prior from PINNs. We finally perform several validation experiments on four different hyperbolic balance laws to highlight the properties of the scheme. Namely, we show that the DG scheme with prior is much more accurate on steady solutions than the DG scheme without prior, while retaining the same approximation quality on unsteady solutions.

Approximately well-balanced Discontinuous Galerkin methods using bases enriched with Physics-Informed Neural Networks

TL;DR

This work introduces a DG method enhanced by basis enrichment with priors obtained from parametric PINNs to achieve approximate well-balancedness for hyperbolic balance laws. By replacing or multiplying basis functions with a learned steady-state prior, the scheme preserves the convergence order while reducing the error constant near steady states, as demonstrated across linear and nonlinear test problems in one and two dimensions. Theoretical results confirm convergence for non-polynomial bases and quantify prior-dependent error bounds, while numerical experiments across linear advection, shallow water, and Euler–Poisson systems show substantial gains in steady-state accuracy and robustness to perturbations. The offline PINN prior construction coupled with online DG basis enrichment yields improved accuracy with modest online costs, suggesting broad applicability and avenues for future extensions such as orthogonalization, neural operators, and time-dependent priors.

Abstract

This work concerns the enrichment of Discontinuous Galerkin (DG) bases, so that the resulting scheme provides a much better approximation of steady solutions to hyperbolic systems of balance laws. The basis enrichment leverages a prior - an approximation of the steady solution - which we propose to compute using a Physics-Informed Neural Network (PINN). To that end, after presenting the classical DG scheme, we show how to enrich its basis with a prior. Convergence results and error estimates follow, in which we prove that the basis with prior does not change the order of convergence, and that the error constant is improved. To construct the prior, we elect to use parametric PINNs, which we introduce, as well as the algorithms to construct a prior from PINNs. We finally perform several validation experiments on four different hyperbolic balance laws to highlight the properties of the scheme. Namely, we show that the DG scheme with prior is much more accurate on steady solutions than the DG scheme without prior, while retaining the same approximation quality on unsteady solutions.
Paper Structure (32 sections, 5 theorems, 113 equations, 4 figures, 21 tables, 2 algorithms)

This paper contains 32 sections, 5 theorems, 113 equations, 4 figures, 21 tables, 2 algorithms.

Table of Contents

  1. Objectives and model
  2. Modified Discontinuous Galerkin scheme
  3. Classical Discontinuous Galerkin scheme
  4. Enrichment of the modal DG basis
  5. Error estimates
  6. Convergence in non-polynomial DG bases
  7. Estimate with prior dependency
  8. Prior construction and algorithm
  9. Parametric PINNs
  10. Algorithm
  11. Applications and numerical results
  12. Linear advection
  13. Study of the stability condition
  14. Steady solution
  15. Perturbed steady solution
  16. ...and 17 more sections

Key Result

Lemma 3.1

Consider an approximation vector space $V_h$ with local basis $(v_{k,0},\dots,v_{k,q})$, which may depend on the cell $\Omega_k$. If there exists constant real numbers $a_{j\ell}$ and $b_j$ independent of the size of the cell $\Delta x_k$ such that, in each cell $\Omega_k$, then for any function $u\in H^{q+1}(\Omega_k)$, there exists $v_h \in V_h$ and a constant real number $C$ independent of $\D

Figures (4)

  • Figure 1: Advection equation: distance between the DG solution $u$ and the underlying steady solution $u_\text{eq}$, with respect to time, for the approximation of a perturbed steady solution for bases with and without prior.
  • Figure 2: Shallow water equations, compactly supported topography: distance between the DG solution $h$ and the underlying steady solution $h_\text{eq}$, with respect to time, for the approximation of a perturbed subcritical steady solution for bases with and without prior.
  • Figure 3: Euler equations, ideal gas pressure law: results for the spherical blast wave. From left to right, we plot the density, the velocity $u = Q / \rho$, and the pressure with respect to space.
  • Figure 4: 2D shallow water equations: representation of the error between the perturbed solution and the underlying steady solution. Left panels: classical basis $V_h$; right panels: prior-enriched basis $V_h^+$. From top to bottom, several final times are displayed ($T = 0.1$, $T = 0.6$, $T = 1.2$).

Theorems & Definitions (9)

  • Lemma 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof