Adam R Stinchcombe

Xinwen Ding, Adam R Stinchcombe

Elliptic interface problems arise in numerous scientific and engineering applications, modeling heterogeneous materials in which physical properties change discontinuously across interfaces. In this paper, we present \textit{Walk-on-Interfaces} (WoI), a grid-free Monte Carlo estimator for a class of Neumann elliptic interface problems with nonhomogeneous flux jump conditions. Our Monte Carlo estimators maintain consistent accuracy throughout the domain and, thus, do not suffer from the well-known close-to-source evaluation issue near the interfaces. We also presented a simple modification with reduced variance. Estimation of the gradient of the solution can be performed, with almost no additional cost, by simply computing the gradient of the Green's function in WoI. Taking a scientific machine learning approach, we use our estimators to provide training data for a deep neural network that outputs a continuous representation of the solution. This regularizes our solution estimates by removing the high-frequency Monte Carlo error. All of our estimators are highly parallelizable, have a $\mathcal{O}(1 / \sqrt{\mathcal{W}})$ convergence rate in the number of samples, and generalize naturally to higher dimensions. We solve problems with many interfaces that have irregular geometry and in up to dimension six. Numerical experiments demonstrate the effectiveness of the approach and to highlight its potential in solving problems motivated by real-world applications.

Jihun Han, Mihai Nica, Adam R Stinchcombe

We introduce a deep neural network based method for solving a class of elliptic partial differential equations. We approximate the solution of the PDE with a deep neural network which is trained under the guidance of a probabilistic representation of the PDE in the spirit of the Feynman-Kac formula. The solution is given by an expectation of a martingale process driven by a Brownian motion. As Brownian walkers explore the domain, the deep neural network is iteratively trained using a form of reinforcement learning. Our method is a 'Derivative-Free Loss Method' since it does not require the explicit calculation of the derivatives of the neural network with respect to the input neurons in order to compute the training loss. The advantages of our method are showcased in a series of test problems: a corner singularity problem, an interface problem, and an application to a chemotaxis population model.