Symbolic Graph Networks for Robust PDE Discovery from Noisy Sparse Data

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.

Discovering governing partial differential equations (PDEs) from noisy observational data is a fundamental challenge in scientific machine learning. Traditional symbolic regression (SR) methods often struggle to identify accurate equations within vast combinatorial search spaces, largely due to their inability to incorporate essential domain-specific prior knowledge. Furthermore, reliance on pointwise evaluations and discrete finite differences inherently amplifies high-frequency noise, creating deceptive fitness landscapes that derail the optimization process. To resolve these bottlenecks, we propose LLM-PDESR, a framework that integrates the structural hypothesis generation of Large Language Models (LLMs) with a mathematically rigorous evaluation environment. By employing C^4-continuous quintic splines for robust differentiation and subdomain weighted residuals as natural low-pass filters, our approach effectively mitigates the fitness landscape distortion that plagues existing methods. A Pareto-driven feedback loop then enables the LLM to iteratively refine candidate equations, balancing predictive accuracy with structural parsimony. We evaluate LLM-PDESR on 23 canonical PDEs and five structurally novel equations (including a multivariate system) specifically designed to preclude dataset memorization and test true discovery capabilities. Demonstrating real-world applicability, the framework successfully extracts a consistent structural skeleton for an interpretable 1D dynamical surrogate (1D-CACE) directly from noisy ERA5 reanalysis data. Extensive experiments and out-of-distribution testing confirm that LLM-PDESR significantly outperforms state-of-the-art methodologies in structural recovery, noise resilience, and the avoidance of spurious complexity and equation bloat.

Models based on partial differential equations (PDEs) are powerful for describing a wide range of complex phenomena in the natural sciences. Accurately identifying the PDE model, which represents the underlying physical law, is essential for a proper understanding of the problem. This reconstruction typically relies on indirect and noisy measurements of the system's state and, without specifically tailored methods, rarely yields symbolic expressions, thereby limiting interpretability. In this work, we address this limitation by considering neural network architectures based on rational functions for the symbolic representation of physical laws. These networks combine the approximation power of rational functions with the flexibility to represent arithmetic operations, and generalize ParFam and EQL-type architectures used in symbolic regression for physical law learning. We further establish regularity results for these symbolic networks. Our main contribution is a reconstruction result showing that, if there exists an admissible physical law that is expressible within the symbolic network architecture, then in the limit of noiseless and complete measurements, symbolic networks recover a physical law within the PDE model that is representable by the architecture. Moreover, the recovered law corresponds to a regularization-minimizing parameterization, promoting interpretability and sparsity in case of $L^1$-regularization. Under an additional identifiability condition, the unique true physical law is recovered. These reconstruction and regularity results are derived at the continuous level prior to discretization due to a formulation in function space. Empirical results using the ParFam architecture are consistent with the theoretical findings and suggest the feasibility of reconstructing interpretable physical laws in practice.