Enhancing Neural Network Differential Equation Solvers
This addresses the problem of numerical solution accuracy for differential equations in computational mathematics, but it is incremental as it builds on existing neural network methods.
The paper tackles solving differential equations using neural networks by proving that feed-forward networks can arbitrarily minimize an objective function for Poisson's equation, ensuring solutions can get arbitrarily close to exact ones, and demonstrates enhancement through error correction networks with numerical experiments on variants of Poisson's equation.
We motivate the use of neural networks for the construction of numerical solutions to differential equations. We prove that there exists a feed-forward neural network that can arbitrarily minimise an objective function that is zero at the solution of Poisson's equation, allowing us to guarantee that neural network solution estimates can get arbitrarily close to the exact solutions. We also show how these estimates can be appreciably enhanced through various strategies, in particular through the construction of error correction networks, for which we propose a general method. We conclude by providing numerical experiments that attest to the validity of all such strategies for variants of Poisson's equation.