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PAS-Net: Physics-informed Adaptive Scale Deep Operator Network

Changhong Mou, Yeyu Zhang, Xuewen Zhu, Qiao Zhuang

TL;DR

PAS-Net introduces a physics-informed adaptive-scale embedding that augments PI-DeepONet with locally rescaled coordinates to better capture multiscale and localized features in nonlinear, singularly perturbed PDEs. The method links this architectural innovation to improved training dynamics via the Neural Tangent Kernel, showing that the adaptive scale increases the NTK's smallest eigenvalue and accelerates convergence. Through numerical experiments on Burgers, diffusion–reaction, and Eikonal equations, PAS-Net consistently achieves higher accuracy and faster convergence than DeepONet and PI-DeepONet under comparable training costs. The work offers a principled multiscale operator-learning framework with potential extensions to inverse problems and more complex PDE systems.

Abstract

Nonlinear physical phenomena often show complex multiscale interactions; motivated by the principles of multiscale modeling in scientific computing, we propose PAS-Net, a physics-informed Adaptive-Scale Deep Operator Network for learning solution operators of nonlinear and singularly perturbed evolution PDEs with small parameters and localized features. Specifically, PAS-Net augments the trunk input in the physics informed Deep Operator Network (PI-DeepONet) with a prescribed (or learnable) locally rescaled coordinate transformation centered at reference points. This addition introduces a multiscale feature embedding that acts as an architecture-independent preconditioner which improves the representation of localized, stiff, and multiscale dynamics. From an optimization perspective, the adaptive-scale embedding in PAS-Net modifies the geometry of the Neural Tangent Kernel (NTK) associated with the neural network by increasing its smallest eigenvalue, which in turn improves spectral conditioning and accelerates gradient-based convergence. We further show that this adaptive-scale mechanism explicitly accelerates neural network training in approximating functions with steep transitions and strong asymptotic behavior, and we provide a rigorous proof of this function-approximation result within the finite-dimensional NTK matrix framework. We test the proposed PAS-Net on three different problems: (i) the one-dimensional viscous Burgers equation, (ii) a nonlinear diffusion-reaction system with sharp spatial gradients, and (iii) a two-dimensional eikonal equation. The numerical results show that PAS-Net consistently achieves higher accuracy and faster convergence than the standard DeepONet and PI-DeepONet models under a similar training cost.

PAS-Net: Physics-informed Adaptive Scale Deep Operator Network

TL;DR

PAS-Net introduces a physics-informed adaptive-scale embedding that augments PI-DeepONet with locally rescaled coordinates to better capture multiscale and localized features in nonlinear, singularly perturbed PDEs. The method links this architectural innovation to improved training dynamics via the Neural Tangent Kernel, showing that the adaptive scale increases the NTK's smallest eigenvalue and accelerates convergence. Through numerical experiments on Burgers, diffusion–reaction, and Eikonal equations, PAS-Net consistently achieves higher accuracy and faster convergence than DeepONet and PI-DeepONet under comparable training costs. The work offers a principled multiscale operator-learning framework with potential extensions to inverse problems and more complex PDE systems.

Abstract

Nonlinear physical phenomena often show complex multiscale interactions; motivated by the principles of multiscale modeling in scientific computing, we propose PAS-Net, a physics-informed Adaptive-Scale Deep Operator Network for learning solution operators of nonlinear and singularly perturbed evolution PDEs with small parameters and localized features. Specifically, PAS-Net augments the trunk input in the physics informed Deep Operator Network (PI-DeepONet) with a prescribed (or learnable) locally rescaled coordinate transformation centered at reference points. This addition introduces a multiscale feature embedding that acts as an architecture-independent preconditioner which improves the representation of localized, stiff, and multiscale dynamics. From an optimization perspective, the adaptive-scale embedding in PAS-Net modifies the geometry of the Neural Tangent Kernel (NTK) associated with the neural network by increasing its smallest eigenvalue, which in turn improves spectral conditioning and accelerates gradient-based convergence. We further show that this adaptive-scale mechanism explicitly accelerates neural network training in approximating functions with steep transitions and strong asymptotic behavior, and we provide a rigorous proof of this function-approximation result within the finite-dimensional NTK matrix framework. We test the proposed PAS-Net on three different problems: (i) the one-dimensional viscous Burgers equation, (ii) a nonlinear diffusion-reaction system with sharp spatial gradients, and (iii) a two-dimensional eikonal equation. The numerical results show that PAS-Net consistently achieves higher accuracy and faster convergence than the standard DeepONet and PI-DeepONet models under a similar training cost.

Paper Structure

This paper contains 23 sections, 3 theorems, 39 equations, 9 figures, 4 tables.

Table of Contents

  1. Introduction
  2. PAS-Net
  3. Partial differential equations with small parameters
  4. Physics-informed Deep Operator Network
  5. Adaptive Scale Embedding
  6. Convergence of Function Approximation with Adaptive Scale
  7. An Example of Function Approximation with Adaptive Scale Feature
  8. PAS-Net Framework
  9. Numerical Results
  10. Burgers equation
  11. Training and testing data
  12. Neural network architectures
  13. Error metric
  14. Diffusion-reaction equations
  15. Training and testing data
  16. ...and 8 more sections

Key Result

Proposition 1

Let $h_\theta:\mathcal{X}\to\mathbb{R}$ be a sufficiently wide neural network trained on a target function $f$ by gradient descent on $\mathcal{L}(\theta)=\tfrac{1}{2}\,\mathbb{E}_{x\sim\mu}[(h_\theta(x)-f(x))^2]$. In the lazy-training regime, i.e. the linear regime, linearization around $\theta_0$ and satisfies the exponential decay where $\lambda_{\min}>0$ is the smallest nonzero eigenvalue of

Figures (9)

  • Figure 1: Illustrations of adaptive-scale features in approximating a steep-gradient function.
  • Figure 2: Illustration of PAS-Net framework.
  • Figure 3: Test problems: (a) Burgers equation, (b) diffusion–reaction equation, and (c) Eikonal equation.
  • Figure 4: Benchmark solutions of three different samples used in Figure \ref{['fig:burgers-result']} for one-dimensional Burgers equation.
  • Figure 5: Spatiotemporal fields of one dimensional Burgers equation with different initial conditions for PI-DeepONet and PAS-Net with different $\gamma$ values.
  • ...and 4 more figures

Theorems & Definitions (5)

  • Proposition 1: NTK-based Convergence for Function Approximation
  • Remark 1: Infinite-Width Limit
  • Proposition 2: NTK-based Convergence for Function Approximation with Adaptive Scale Feature
  • Proposition 3: Matrix monotonicity of adaptive scale NTK
  • proof