Moving Sampling Physics-informed Neural Networks induced by Moving Mesh PDE

TL;DR

The paper tackles the sampling bottleneck in physics-informed neural networks (PINN) by introducing MMPDE-Net, a moving-mesh PDE–driven adaptive sampler that relocates points to regions of high solution variation without altering mesh topology. It then couples this sampler with PINN to form MS-PINN, using a three-stage workflow (pre-train PINN, adapt sampling via MMPDE-Net, then formal PINN training with transferred parameters) and proves an error bound indicating potential improvements over standard PINN. Theoretical analysis based on weighted norms and Rademacher complexity shows MS-PINN can reduce approximation error with high probability when sampling concentrates where the residual is large. Empirically, MS-PINN demonstrates superior accuracy across 2D Poisson problems (one and two peaks) and Burgers equations (forward and inverse, 1D and 2D) compared with PINN and several adaptive baselines, validating the efficacy of solver-independent adaptive sampling for complex PDEs.

Abstract

In this work, we propose an end-to-end adaptive sampling neural network (MMPDE-Net) based on the moving mesh method, which can adaptively generate new sampling points by solving the moving mesh PDE. This model focuses on improving the quality of sampling points generation. Moreover, we develop an iterative algorithm based on MMPDE-Net, which makes the sampling points more precise and controllable. Since MMPDE-Net is a framework independent of the deep learning solver, we combine it with physics-informed neural networks (PINN) to propose moving sampling PINN (MS-PINN) and demonstrate its effectiveness by error analysis under some assumptions. Finally, we demonstrate the performance improvement of MS-PINN compared to PINN through numerical experiments of four typical examples, which numerically verify the effectiveness of our method.

Figures (35)

• Figure 1: Variations of the minimum of the loss function and the relative $L_2$ error with different number of sampling points ($M_{r2}$ =100,150,250,300) for $k=16$.
• Figure 2: Flow chart of MMPDE-Net.
• Figure 3: Points distribution for $u=ce^{-c^2{(x^2+y^2)}}$ and $w = (1+u^2)^{\frac{1}{2}}$ in 2D case when $c$ is increased. (a) $c =5$; (b) $c =10$; (c) $c =50$; (d) $c =100$.
• Figure 4: (a) the gradient of $u = e^{-(4x^2 + 9y^2 - 1)^2}$; (b) points distribution with $w = (1+(\lvert \nabla u \rvert)^2)^{\frac{1}{2}}$.
• Figure 5: Points distribution for $u=50e^{-c^2{(x^2+y^2)}}$ and $w = (1+u^2)^{\frac{1}{2}}$ after iterations. (a) after 1 iteration; (b) after 2 iterations; (c) after 3 iterations; (d) after 5 iterations.

Theorems & Definitions (13)

• Definition 1
• Remark 1
• Theorem 1
• proof
• Definition 2
• Lemma 1
• Theorem 2
• proof
• Corollary 1
• proof