Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operators

Partial differential equations (PDEs) are central to scientific modeling. Modern workflows increasingly rely on learning-based components to support model reuse, inference, and integration across large computational processes. Despite the emergence of various physics-aware data-driven approaches, the field still lacks a unified perspective to uncover their relationships, limitations, and appropriate roles in scientific workflows. To this end, we propose a unifying perspective to place two dominant paradigms: Physics-Informed Neural Networks (PINNs) and Neural Operators (NOs), within a shared design space. We organize existing methods from three fundamental dimensions: what is learned, how physical structures are integrated into the learning process, and how the computational load is amortized across problem instances. In this way, many challenges can be best understood as consequences of these structural properties of learning PDEs. By analyzing advances through this unifying view, our survey aims to facilitate the development of reliable learning-based PDE solvers and catalyze a synthesis of physics and data.

💡 Research Summary

This survey presents a unified perspective on two dominant learning‑based approaches for solving partial differential equations (PDEs): Physics‑Informed Neural Networks (PINNs) and Neural Operators (NOs). By framing both methods within a common design space, the authors identify three fundamental structural dimensions that differentiate them: (T1) the object of learning—instance‑specific solution fields for PINNs versus family‑level operators mapping problem specifications to solutions for NOs; (T2) the supervision interface—physics‑as‑supervision for PINNs versus paired data supervision (with physics injected through architecture, regularization, or diagnostics) for NOs; and (T3) the locus of amortization—per‑instance optimization for PINNs versus offline training with amortized inference for NOs.

The paper argues that these structural choices are not merely algorithmic preferences but are dictated by the “workflow contract” in which a PDE solver is embedded. Such contracts specify permissible perturbations at deployment (e.g., changes in coefficients, geometry, discretization, observation operators, or temporal horizon) and the required reliability signals (boundary feasibility, conservation, uncertainty quantification, out‑of‑distribution detection, fallback mechanisms).

Three persistent challenges arise across all learning‑based PDE solvers: (C1) multiscale representation, (C2) geometry and boundary feasibility, and (C3) trustworthiness under contract shift. The authors detail how PINNs, which treat the PDE residual as a primary loss term, often struggle with stiff multi‑objective optimization, leading to poor high‑frequency learning and boundary violations. Neural Operators, which rely on large paired datasets, are vulnerable to distribution shift, discretization mismatch, and geometry changes, making them less robust when the deployment domain differs from training.

To address these issues, the survey catalogs recent advances and hybrid strategies. For PINNs, developments include adaptive weighting of loss terms, multi‑scale loss functions, Fourier or graph feature embeddings, and physics‑injection layers that improve representation of fine‑scale structures. For Neural Operators, innovations such as DeepONet, Fourier Neural Operators, Wavelet Neural Operators, and Graph Neural Operators expand expressive power, while physics‑informed regularizers, architecture‑level conservation constraints, and post‑hoc refinement improve physical fidelity. Hybrid approaches (e.g., PI‑NO, PI‑MF) combine residual‑based supervision with operator learning to simultaneously mitigate C1‑C3.

The authors emphasize that evaluation must go beyond average L2 error. Meaningful metrics should align with the workflow contract: spectral or band‑limited error, boundary violation measures, conservation checks, rollout stability, calibration of probabilistic outputs, out‑of‑distribution detection, and the ability to trigger refinement or fallback under stress tests.

Finally, the survey outlines open research directions: establishing standardized contracts and benchmark suites, integrating multimodal observations, developing formal verification and uncertainty propagation techniques, and building end‑to‑end pipelines that guarantee reproducibility, scalability, and reliability. By situating PINNs and Neural Operators within a shared taxonomy, the paper provides a roadmap for the systematic development of trustworthy, data‑efficient PDE solvers that can be seamlessly incorporated into scientific computing workflows.