Hybrid Numerical–Machine Learning Techniques for Solving Multi-Physics Nonlinear Partial Differential Equations

Source abstract

Multi-physics nonlinear partial differential equations (PDEs) govern a wide spectrum of complex systems in engineering, physics, and applied sciences, including fluid–structure interaction, heat transfer with chemical reactions, and electromagnetic–thermal coupling. Classical numerical methods such as finite difference, finite element and finite volume methods have been used extensively in a bid to estimate solutions of such equations though are commonly computationally intensive, face stability issues and also to address highly nonlinear and coupled problems. New opportunities on how numerical solvers can be optimized with the help of data-driven approximations and adaptive learning have been also developed in recent developments in machine learning, in particular, deep learning. The paper will describe an overall architecture of hybrid numerical-machine learning systems balancing between classical discretization algorithms and neural network-based surrogacy and physics-informed learning. Combination methodology A deterministic solver coupled with deep neural network to increase convergence, accuracy and computational efficiency in solving solutions to the multi-physics nonlinear PDEs. Benchmarking is done on problems in the coupled heat transfer and fluid flow, reaction-diffusion system, and nonlinear elasticity. The results demonstrate that hybrid approaches can be beneficial in achieving a significant reduction in the level of computational complexity but with maintaining good fidelity in the model of nonlinear interactions. The article offers a general and effective paradigm of the solution of the complicated PDE models and describes the promise of hybrid structures to scientific computing.