Numerical Analysis of Evolutionary Equations with Multivalued Interaction Functions

This paper serves as a continuation of previous research on approximating solutions for reaction-diffusion equations with multivalued interaction functions. While previous work established a Machine Learning framework using Physics-Informed Neural Networks (PINNs) and Pasch-Hausdorffregularization, this study investigates the underlying Monte Carlo integration techniques required to estimate the loss functionals for such non-unique, discontinuous problems. We analyze the convergence properties of Monte Carlo estimators applied to the regularized interaction terms and demonstrate that as the regularization parameter k → ∞, the variance of the standard Monte Carlo estimator grows unboundedly, necessitating advanced sampling strategies. We propose an Adaptive Importance Sampling approach to improve the accuracy of stochastic optimization in the presence of multivalued terms and provide rigorous error bounds for the proposed method.