Reliable Scientific Machine Learning for PDE Solution and Discovery

Partial differential equations (PDEs) are central to the mathematical modeling of physical, chemical, biological, and engineering systems, including fluid flow, heat transfer, electromagnetism, and structural dynamics. Established numerical methods such as spectral, finite-difference, finite-element, and finite-volume methods provide reliable tools for many forward problems, but can become computationally expensive for high-dimensional, multiscale, stiff, and multiphysics systems. They also face challenges when sparse, noisy, or incomplete observations must be incorporated into inverse problems.Scientific machine learning has emerged as a complementary approach by integrating physical laws with data-driven models. Physics-informed neural networks (PINNs) and neural operators have demonstrated potential for mesh-free approximation, parameter identification, data assimilation, and rapid surrogate modeling. However, questions remain regarding their optimization, convergence, stability, generalization, interpretability, and reliability, particularly for problems involving shocks, high gradients, non-local operators, and complex geometries. Further investigation is needed to establish when these methods outperform or complement conventional solvers and how they can be developed into dependable scientific computing tools.This Research Topic aims to advance the theory, methodology, and application of PINNs and related operator-learning approaches for forward and inverse PDE problems. It will examine how network architectures, training strategies, physical constraints, and numerical formulations can improve accuracy, stability, computational efficiency, and robustness. Contributions may address theoretical convergence and error estimates, optimization landscapes, spectral bias, uncertainty quantification, and the treatment of sparse or noisy data. The Research Topic also seeks to clarify the comparative advantages and limitations of PINNs and neural operators relative to traditional numerical methods, while promoting interpretable and reproducible approaches for complex scientific and engineering applications.To gather further insights into physics-informed and operator-learning methods for PDEs, this Research Topic will focus on methodological developments, theoretical analysis, inverse modeling, surrogate construction, and representative applications. We welcome articles addressing, but not limited to, the following themes:- PINN architectures, including variational, fractional, conservative, transformer-based, and causality-aware formulations- Adaptive activation functions, domain decomposition, localized representations, and parallel training strategies- Adaptive loss weighting, multi-objective optimization, transfer learning, and pretrained models for PINNs- Optimization landscapes, spectral bias, convergence estimates, error bounds, stability, and generalization theory- Numerical treatment of stiff systems, shocks, high-gradient regions, multiscale behavior, and non-local PDEs- Parameter estimation, system identification, and discovery of sources, boundary conditions, or initial conditions- Bayesian PINNs, uncertainty quantification, data assimilation, and real-time process control- Neural operators, including DeepONets, Fourier Neural Operators, and conditional operator-learning frameworks- Neural fields, generative models, and efficient surrogate modeling for spatiotemporal scientific systems- Applications in fluid mechanics, aerodynamics, multiphase flow, structural mechanics, wave propagation, and non-destructive evaluation- Applications in oncology, tissue mechanics, physiological modeling, geophysics, groundwater flow, heat transfer, combustion, smart grids, and electromagnetism Topic editors: Shina Oloniiju, Precious Sibanda, Afrah Farea, Saiful Khan Source: https://www.frontiersin.org/research-topics/84405/reliable-scientific-machine-learning-for-pde-solution-and-discovery