Adversarial Sampling for Solving Differential Equations with Neural Networks
This work addresses a specific bottleneck in computational methods for researchers and engineers, representing an incremental improvement in sampling techniques.
The paper tackles the problem of inefficient sampling in neural network-based differential equation solvers by introducing an adversarial sampling scheme that selects points to maximize the current loss, and demonstrates that this approach outperforms existing methods on various problems.
Neural network-based methods for solving differential equations have been gaining traction. They work by improving the differential equation residuals of a neural network on a sample of points in each iteration. However, most of them employ standard sampling schemes like uniform or perturbing equally spaced points. We present a novel sampling scheme which samples points adversarially to maximize the loss of the current solution estimate. A sampler architecture is described along with the loss terms used for training. Finally, we demonstrate that this scheme outperforms pre-existing schemes by comparing both on a number of problems.