DMIS: Dynamic Mesh-based Importance Sampling for Training Physics-Informed Neural Networks

TL;DR

This paper tackles the inefficiencies and stability issues in training physics-informed neural networks (PINNs) by introducing Dynamic Mesh-based Importance Sampling (DMIS). DMIS integrates importance sampling into mini-batch PDE residual sampling and uses Dynamic Mesh-based Weight Estimation (DMWE) with a Delaunay triangulation to efficiently estimate per-point weights, updating the mesh according to the loss landscape. Across Schrödinger, Burgers', and KdV benchmarks, DMIS consistently improves convergence speed and prediction accuracy relative to state-of-the-art PINN baselines, with notable gains on nonlinear PDEs like the Schrödinger equation. The approach is general, computationally light, and demonstrates strong generalization across PINN architectures, making DMIS a practical tool for accelerating physics-informed learning.

Abstract

Modeling dynamics in the form of partial differential equations (PDEs) is an effectual way to understand real-world physics processes. For complex physics systems, analytical solutions are not available and numerical solutions are widely-used. However, traditional numerical algorithms are computationally expensive and challenging in handling multiphysics systems. Recently, using neural networks to solve PDEs has made significant progress, called physics-informed neural networks (PINNs). PINNs encode physical laws into neural networks and learn the continuous solutions of PDEs. For the training of PINNs, existing methods suffer from the problems of inefficiency and unstable convergence, since the PDE residuals require calculating automatic differentiation. In this paper, we propose Dynamic Mesh-based Importance Sampling (DMIS) to tackle these problems. DMIS is a novel sampling scheme based on importance sampling, which constructs a dynamic triangular mesh to estimate sample weights efficiently. DMIS has broad applicability and can be easily integrated into existing methods. The evaluation of DMIS on three widely-used benchmarks shows that DMIS improves the convergence speed and accuracy in the meantime. Especially in solving the highly nonlinear Schrödinger Equation, compared with state-of-the-art methods, DMIS shows up to 46% smaller root mean square error and five times faster convergence speed. Code are available at https://github.com/MatrixBrain/DMIS.

Figures (8)

• Figure 1: Comparison with stat-of-the-art methods (PINN-O, PINN-N). DMIS achieves faster convergence speed and better prediction accuracy.
• Figure 2: The Training of PINNs with DMIS. In each iteration, sample weights are estimated by DMWE, and a mini-batch for PDE residuals is sampled by DMIS. The derivatives terms are computed by automatic differentiation. These mini-batches are applied to compute the initial conditions loss $\mathcal{L}_i$, boundary conditions loss $\mathcal{L}_b$, and PDE residuals loss $\mathcal{L}_f$. DMIS reweights sample weights of $M_f$ and conveys these weights to the loss function $\mathcal{L}$. The mesh of DMIS is updated according to $\mathcal{L}$.
• Figure 3: The Schrödinger Equation. (a, b, c) The prediction errors. (d) Convergence curves. (e, f, g) show the prediction (red) and ground-truth (blue).
• Figure 4: The Burgers' Equation. (a, b, c) The prediction errors. (d) Convergence curves. (e, f, g) show the prediction (red) and ground-truth (blue).
• Figure 5: The KdV Equation. (a, b, c) The prediction errors. (d) Convergence curves. (e, f, g) show the prediction (red) and ground-truth (blue).