Scientific Machine Learning for PDEs: Operators, Surrogates, and Error-Controlled Multi-Fidelity Schemes

Abstract

This review aims to synthesize recent advances in scientific machine learning (SciML) for solving partial differential equations (PDEs), focusing on operator learning, surrogate modeling, and error-controlled multi-fidelity frameworks that integrate data-driven intelligence with physical consistency. This study adopted a qualitative, interpretive review design based on a systematic literature analysis of thirteen peer-reviewed journal articles published between 2019 and 2025. The data collection process relied exclusively on scholarly databases such as Scopus, ScienceDirect, and IEEE Xplore, targeting works addressing neural operator architectures, hybrid physics–ML couplings, and multi-fidelity adaptation. All sources were imported into Nvivo 14 software for coding and thematic synthesis. Open, axial, and selective coding cycles were performed until theoretical saturation was achieved. Four main categories—operator learning paradigms, surrogate and reduced-order models, error-controlled multi-fidelity schemes, and computational integration—were extracted and structured according to their conceptual relationships and methodological contributions. The review identified that operator learning (e.g., DeepONet, Fourier Neural Operator, and physics-informed variants) provides a scalable framework for learning function-to-function mappings across PDE families. Surrogate modeling emerged as an efficient approach for reduced-order representation and hybrid PDE–ML coupling, while sparse, compressive, and latent-space techniques improved model interpretability and efficiency. Multi-fidelity architectures, integrating uncertainty quantification and adaptive refinement, offered robust mechanisms for cost-accuracy optimization and error control. Finally, the implementation trend emphasized high-performance computing, benchmarking (PDEBench), hybrid symbolic–numeric integration, and reproducibility practices as essential to operational deployment. Scientific machine learning for PDEs is transitioning from experimental novelty to a mature computational paradigm that unifies physics-informed theory, data-driven surrogacy, and adaptive error control. Its promise lies in producing generalizable, trustworthy, and computationally efficient solvers that can accelerate discovery across domains such as fluid mechanics, climate modeling, and structural dynamics while maintaining physical interpretability and numerical rigor.