Physically Misinformed Neural Networks: Evaluating PINN Assumptions, with Applications to European Option Pricing

Abstract

In recent years, Physics Informed Neural Networks (PINNs) have emerged as a powerful technique for incorporating domain knowledge into the machine learning modeling process. Specifically, the modeler trains a neural network on observed data while simultaneously penalizing deviations in the learned function from a set of posited PDE conditions. Understandably, if the modeler unwittingly enforces PDE constraints that poorly describe a given problem, this may significantly hinder the model’s generalizability to unseen data. The purpose of this paper is therefore to design a hypothesis test to evaluate the likelihood of a PINN formulation, when enforcing a chosen PDE prior, to improve fit over a baseline physically-uninformed neural network. We design a novel approach, using conformal prediction techniques, and outline conditions under which our algorithm can test null hypotheses that quantify the expected performance of the PINN. We justify the hypothesis test with simulated data generated to adhere to the Heat Equation, showing that our test functions as expected. Then, we show how to employ our hypothesis test to conclude that the Black Scholes equation is a useful regularizer within a PINN framework for European call option pricing, an important potential application of our work to a regime with an unknown data governing function.