1 research outputs found
The Neural Network Approach to Inverse Problems in Differential Equations
We proposed a framework for solving inverse problems in differential
equations based on neural networks and automatic differentiation. Neural
networks are used to approximate hidden fields. We analyze the source of errors
in the framework and derive an error estimate for a model diffusion equation
problem. Besides, we propose a way for sensitivity analysis, utilizing the
automatic differentiation mechanism embedded in the framework. It frees people
from the tedious and error-prone process of deriving the gradients. Numerical
examples exhibit consistency with the convergence analysis and error saturation
is noteworthily predicted. We also demonstrate the unique benefits neural
networks offer at the same time: universal approximation ability, regularizing
the solution, bypassing the curse of dimensionality and leveraging efficient
computing frameworks.Comment: 32 pages, 9 figure