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

• URL: http://arxiv.org/abs/2603.22951v1
• Date: Tue, 24 Mar 2026 08:47:01 GMT
• Title: Weak-PDE-Net: Discovering Open-Form PDEs via Differentiable Symbolic Networks and Weak Formulation
• Abstract summary: Weak-PDE-Net is an end-to-end differentiable framework that can robustly identify open-form PDEs.<n>Experiments on several challenging PDE benchmarks demonstrate that Weak-PDE-Net accurately recovers governing equations, even under highly sparse and noisy observations.
• Score: 10.638660239419636
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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