| Abstract: |
| Traditional numerical methods are mathematically rigorous, highly accurate, and physically conservative, forming the reliable foundation of modern scientific computing. Despite their success, they still face intrinsic challenges such as the need for complex mesh generation, limited flexibility in representing global structures, case-by-case reassembly for new configurations, the curse of dimensionality, and difficulties in integrating data or uncertainty. Neural network-based numerical methods have recently emerged as an alternative paradigm, aiming to overcome these limitations through their strong representation capability. However, traditional training-based neural approaches often suffer from the challenges of nonlinear, nonconvex optimization, resulting in limited accuracy and efficiency. To address these issues, we propose a family of randomized neural network (RaNN) methods that integrate the mathematical rigor of classical numerical schemes with the flexibility of neural representations. The framework includes the RaNN-Petrov-Galerkin (RaNN-PG), Local RaNN-Discontinuous Galerkin (LRaNN-DG), LRaNN-HDPG, and LRaNN-Finite Difference methods. Furthermore, an Adaptive-Growth RaNN (AG-RaNN) strategy is introduced to optimize the weighting and selection of random parameters dynamically. We also explore RaNNs for accelerating operator learning, enabling faster training for parameterized PDEs. Numerical experiments demonstrate that RaNN-based methods achieve high accuracy with few degrees of freedom while remaining mesh-free, structure-preserving, and easily extendable to high-dimensional and time-dependent problems. These results highlight RaNNs as a promising direction for unifying classical numerical methods and modern machine learning in the efficient solution of PDEs. |
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