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Weak-PDE-Net: Discovering Open-Form PDEs via Differentiable Symbolic Networks and Weak Formulation

Xinxin Li, Xingyu Cui, Jin Qi, Juan Zhang, Da Li, Junping Yin

Abstract

Discovering governing Partial Differential Equations (PDEs) from sparse and noisy data is a challenging issue in data-driven scientific computing. Conventional sparse regression methods often suffer from two major limitations: (i) the instability of numerical differentiation under sparse and noisy data, and (ii) the restricted flexibility of a pre-defined candidate library. We propose Weak-PDE-Net, an end-to-end differentiable framework that can robustly identify open-form PDEs. Weak-PDE-Net consists of two interconnected modules: a forward response learner and a weak-form PDE generator. The learner embeds learnable Gaussian kernels within a lightweight MLP, serving as a surrogate model that adaptively captures system dynamics from sparse observations. Meanwhile, the generator integrates a symbolic network with an integral module to construct weak-form PDEs, avoiding explicit numerical differentiation and improving robustness to noise. To relax the constraints of the pre-defined library, we leverage Differentiable Neural Architecture Search strategy during training to explore the functional space, which enables the efficient discovery of open-form PDEs. The capability of Weak-PDE-Net in multivariable systems discovery is further enhanced by incorporating Galilean Invariance constraints and symmetry equivariance hypotheses to ensure physical consistency. Experiments on several challenging PDE benchmarks demonstrate that Weak-PDE-Net accurately recovers governing equations, even under highly sparse and noisy observations.

Weak-PDE-Net: Discovering Open-Form PDEs via Differentiable Symbolic Networks and Weak Formulation

Abstract

Discovering governing Partial Differential Equations (PDEs) from sparse and noisy data is a challenging issue in data-driven scientific computing. Conventional sparse regression methods often suffer from two major limitations: (i) the instability of numerical differentiation under sparse and noisy data, and (ii) the restricted flexibility of a pre-defined candidate library. We propose Weak-PDE-Net, an end-to-end differentiable framework that can robustly identify open-form PDEs. Weak-PDE-Net consists of two interconnected modules: a forward response learner and a weak-form PDE generator. The learner embeds learnable Gaussian kernels within a lightweight MLP, serving as a surrogate model that adaptively captures system dynamics from sparse observations. Meanwhile, the generator integrates a symbolic network with an integral module to construct weak-form PDEs, avoiding explicit numerical differentiation and improving robustness to noise. To relax the constraints of the pre-defined library, we leverage Differentiable Neural Architecture Search strategy during training to explore the functional space, which enables the efficient discovery of open-form PDEs. The capability of Weak-PDE-Net in multivariable systems discovery is further enhanced by incorporating Galilean Invariance constraints and symmetry equivariance hypotheses to ensure physical consistency. Experiments on several challenging PDE benchmarks demonstrate that Weak-PDE-Net accurately recovers governing equations, even under highly sparse and noisy observations.
Paper Structure (53 sections, 58 equations, 4 figures, 15 tables, 1 algorithm)

This paper contains 53 sections, 58 equations, 4 figures, 15 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. System Response Function
  4. Sparse and Noisy Observation Data
  5. Open-form PDE
  6. Optimization Objective
  7. Related Works
  8. Robust PDE Discovery from Sparse and Noisy Data
  9. Open-form PDE Discovery via Symbolic Regression
  10. Search Space Refinement via Physical Consistency
  11. Methodology
  12. The Derivation of the Weak-form PDE
  13. The Selection of Test Functions
  14. The Architecture of Weak-PDE-Net
  15. Learnable Embedding
  16. ...and 38 more sections

Figures (4)

  • Figure 1: The overall architecture of Weak-PDE-Net.
  • Figure 2: Overview of the training process, exemplified by the Burgers equation. The training progress through two parallel tracks. The PDE Learning Process focuses on learning the PDE solution, consistently reconstructing the full grid data. Concurrently, the PDE Discovery Process evolves from symbolic network architecture adaptation, to connectivity pruning for structural sparsity, and finally to precise coefficient refinement. These two tracks are jointly optimized via a shared loss.
  • Figure 3: The forward propagation of the symbolic network during the searching phase. The left panel illustrates the overall forward propagation framework; the right panel provides the specific details of the functional operation.
  • Figure 4: The results of the ablation studies. (a) A radar chart illustrating the True Positive Rate $\text{TPR}$. (b) and (c) Line plots showcasing the parameter errors $E_{\infty}$ and $E_{2}$, respectively. (d) A bar chart representing the reconstructed error $\mathcal{L}(\hat{U},U)$.

Theorems & Definitions (1)

  • Definition 1: Symbolic Network Architecture