Integrating neural networks with numerical schemes for dynamical systems: A review

Abstract

As scientific discovery becomes increasingly data-driven, integrating physics-based numerical methods with advanced machine learning (ML) techniques has brought new insight in the analysis of complex physical systems. This paper explores how this integrated approach overcomes the limitations of traditional first-principle methods and brute-force ML techniques to achieve a more precise solution to complex physical problems. Specifically, we review networks that combine classical numerical schemes with neural networks applied to various physical systems. These integrated methods with residual structures effectively adhere to system symmetries and conservation laws. This integration outperforms conventional data-driven techniques in robustness and predictive capability, even with smaller datasets, owing to its improved ability to capture complex physical patterns.