PDE Discovery

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.

Data-driven discovery of partial differential equations (PDEs) offers a promising paradigm for uncovering governing physical laws from observational data. However, in practical scenarios, measurements are often contaminated by noise and limited by sparse sampling, which poses significant challenges to existing approaches based on numerical differentiation or integral formulations. In this work, we propose a Symbolic Graph Network (SGN) framework for PDE discovery under noisy and sparse conditions. Instead of relying on local differential approximations, SGN leverages graph message passing to model spatial interactions, providing a non-local representation that is less sensitive to high frequency noise. Based on this representation, the learned latent features are further processed by a symbolic regression module to extract interpretable mathematical expressions. We evaluate the proposed method on several benchmark systems, including the wave equation, convection-diffusion equation, and incompressible Navier-Stokes equations. Experimental results show that SGN can recover meaningful governing relations or solution forms under varying noise levels, and demonstrates improved robustness compared to baseline methods in sparse and noisy settings. These results suggest that combining graph-based representations with symbolic regression provides a viable direction for robust data-driven discovery of physical laws from imperfect observations. The code is available at https://github.com/CXY0112/SGN

Generalizing across disparate physical laws remains a fundamental challenge for artificial intelligence in science. Existing deep-learning solvers are largely confined to single-equation settings, limiting transfer across physical regimes and inference tasks. Here we introduce pADAM, a unified generative framework that learns a shared probabilistic prior across heterogeneous partial differential equation families. Through a learned joint distribution of system states and, where applicable, physical parameters, pADAM supports forward prediction and inverse inference within a single architecture without retraining. Across benchmarks ranging from scalar diffusion to nonlinear Navier--Stokes equations, pADAM achieves accurate inference even under sparse observations. Combined with conformal prediction, it also provides reliable uncertainty quantification with coverage guarantees. In addition, pADAM performs probabilistic model selection from only two sparse snapshots, identifying governing laws through its learned generative representation. These results highlight the potential of generative multi-physics modeling for unified and uncertainty-aware scientific inference.

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.

We study operator relevance in data-driven partial differential equation (PDE) discovery. Sparse residual methods can select terms that improve residual fit, but residual contribution is not the same as functional necessity. We formalize this distinction through counterfactual operator interventions, where a candidate term is deleted or perturbed and the factual and intervened trajectories, or observables, are compared. The resulting theory gives six reusable results. A residual--counterfactual gap theorem shows that deletion effects are governed by the inverse linearized PDE map, not by residual magnitude alone. A certified decision theorem gives error margins for relevance, irrelevance, and abstention under neural or numerical surrogate error. An aliasing theorem characterizes experiment-dependent non-identifiability through the null space of the operator-evaluation design. A constraint-manifold theorem shows that operators vanishing on invariant constraint classes cannot be identified from trajectories restricted to those classes. A pruning-consistency theorem proves that sparse screening followed by counterfactual deletion recovers the functionally relevant support under a recall and margin condition. An observable-level adjoint theorem extends relevance testing from full-state deviations to scientific quantities of interest. Validation experiments test these mechanisms on synthetic PDEs with known support and on public geophysical fields from atmospheric reanalysis and NOAA OISST. The real-data results are reported as operator-surrogate diagnostics, not as unconditional recovery of physical laws. The framework provides a rigorous diagnostic layer for distinguishing residual usefulness from counterfactual operator relevance within a specified library, experiment class, norm, and tolerance.

Discovering governing equations from data remains challenging when the underlying dynamics involve nonlocal differential operators, field interactions governed by auxiliary equations, or temporal memory effects. We propose Neural Operator-based symbolic Model approximaTion and discOvery (NOMTO), a framework that extends Equation Learner-type symbolic architectures by incorporating pretrained neural operators as nodes in the symbolic network. NOMTO represents candidate equations as sparse differentiable computational graphs that combine algebraic operations with fixed neural operator surrogates pretrained to approximate nonlinear operators. We evaluate the method on model-discovery problems involving nonlocal spatial operators, couplings mediated by auxiliary field equations, and temporal integral terms representing memory effects. The results show that NOMTO can recover compact governing equations containing nonlocal operator terms, thereby extending symbolic model discovery beyond libraries restricted to local derivatives and point-wise algebraic combinations.