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NewPINNs: Physics-Informing Neural Networks Using Conventional Solvers for Partial Differential Equations

Maedeh Makki, Satish Chandran, Maziar Raissi, Adrien Grenier, Behzad Mohebbi

TL;DR

By delegating the enforcement of physics, boundary conditions, and numerical stability to established numerical solvers, NewPINNs mitigates several well-known failure modes of standard physics-informed neural networks, including optimization pathologies, sensitivity to loss weighting, and poor performance in stiff or nonlinear regimes.

Abstract

We introduce NewPINNs, a physics-informing learning framework that couples neural networks with conventional numerical solvers for solving differential equations. Rather than enforcing governing equations and boundary conditions through residual-based loss terms, NewPINNs integrates the solver directly into the training loop and defines learning objectives through solver-consistency. The neural network produces candidate solution states that are advanced by the numerical solver, and training minimizes the discrepancy between the network prediction and the solver-evolved state. This pull-push interaction enables the network to learn physically admissible solutions through repeated exposure to the solver's action, without requiring problem-specific loss engineering or explicit evaluation of differential equation residuals. By delegating the enforcement of physics, boundary conditions, and numerical stability to established numerical solvers, NewPINNs mitigates several well-known failure modes of standard physics-informed neural networks, including optimization pathologies, sensitivity to loss weighting, and poor performance in stiff or nonlinear regimes. We demonstrate the effectiveness of the proposed approach across multiple forward and inverse problems involving finite volume, finite element, and spectral solvers.

NewPINNs: Physics-Informing Neural Networks Using Conventional Solvers for Partial Differential Equations

TL;DR

By delegating the enforcement of physics, boundary conditions, and numerical stability to established numerical solvers, NewPINNs mitigates several well-known failure modes of standard physics-informed neural networks, including optimization pathologies, sensitivity to loss weighting, and poor performance in stiff or nonlinear regimes.

Abstract

We introduce NewPINNs, a physics-informing learning framework that couples neural networks with conventional numerical solvers for solving differential equations. Rather than enforcing governing equations and boundary conditions through residual-based loss terms, NewPINNs integrates the solver directly into the training loop and defines learning objectives through solver-consistency. The neural network produces candidate solution states that are advanced by the numerical solver, and training minimizes the discrepancy between the network prediction and the solver-evolved state. This pull-push interaction enables the network to learn physically admissible solutions through repeated exposure to the solver's action, without requiring problem-specific loss engineering or explicit evaluation of differential equation residuals. By delegating the enforcement of physics, boundary conditions, and numerical stability to established numerical solvers, NewPINNs mitigates several well-known failure modes of standard physics-informed neural networks, including optimization pathologies, sensitivity to loss weighting, and poor performance in stiff or nonlinear regimes. We demonstrate the effectiveness of the proposed approach across multiple forward and inverse problems involving finite volume, finite element, and spectral solvers.
Paper Structure (65 sections, 2 theorems, 82 equations, 14 figures, 5 tables, 3 algorithms)

This paper contains 65 sections, 2 theorems, 82 equations, 14 figures, 5 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Physics-Informed Neural Networks (PINNs):
  3. Hybrid Neural-Numerical Models:
  4. Data-Driven Discovery of Differential Equations:
  5. Methodology
  6. Physics-Informed Neural Networks (PINNs)
  7. NewPINNs: Physics-Informing Neural Networks using Numerical Solvers
  8. Forward Problems
  9. NewPINNs for Transient Systems:
  10. NewPINNs for Steady-State Systems:
  11. Inverse Problems
  12. Relation to Distillation and Solver-Coupled Learning
  13. Solver-Consistency Guarantees for NewPINNs
  14. Steady-State NewPINNs.
  15. Transient NewPINNs.
  16. ...and 50 more sections

Key Result

Proposition 1

Assume that a steady-state solution $u_\alpha^\star$ exists and that the solver operator $T_\alpha$ is locally contractive in a neighborhood $\mathcal{U}$ of $u_\alpha^\star$, i.e. there exists a constant $q \in (0,1)$ such that Then for any network prediction $u = f_{\mathrm{NN}}(\alpha;\theta) \in \mathcal{U}$, the NewPINNs steady-state residual provides an explicit bound on the distance to eq

Figures (14)

  • Figure 1: Transient NewPINNs: At each training iteration, the neural network produces a candidate solution at time $t_n$ which is advanced to $t_{n+1}$ by the numerical solver. A solver-consistency loss compares this solver-evolved state with the network’s prediction at the next time step, enforcing temporal consistency under solver evolution. The solver is treated as a black-box operator and does not participate in gradient computation.
  • Figure 2: Predicted and true equilibrium solutions (left) and pointwise error (right) for $V(x,y)=1.5\sin(2\pi x)\sin(2\pi y)$.
  • Figure 3: Predicted and true steady-state velocity fields (left) and streamlines (right) for $Re=2500$.
  • Figure 4: Allen--Cahn equation with $\alpha=10^{-3}$. Left: Neural network prediction over the full space--time domain. Right: Network prediction at $t=0.2$ with training points highlighted, and comparison between the network and solver solutions at $t=0.65$.
  • Figure 5: Kuramoto--Sivashinsky equation with $\alpha=1.1$. Left: Neural network prediction over the full space--time domain. Right: Comparison between the neural network and solver solutions at $t=0.1$ and $t=8$.
  • ...and 9 more figures

Theorems & Definitions (4)

  • Proposition 1: A posteriori control via fixed-point residual
  • proof
  • Proposition 2: Solver-consistency stability for transient systems
  • proof