Blake Bullwinkel

Blake Bullwinkel, Dylan Randle, Pavlos Protopapas et al. · harvard

Solutions to differential equations are of significant scientific and engineering relevance. Physics-Informed Neural Networks (PINNs) have emerged as a promising method for solving differential equations, but they lack a theoretical justification for the use of any particular loss function. This work presents Differential Equation GAN (DEQGAN), a novel method for solving differential equations using generative adversarial networks to "learn the loss function" for optimizing the neural network. Presenting results on a suite of twelve ordinary and partial differential equations, including the nonlinear Burgers', Allen-Cahn, Hamilton, and modified Einstein's gravity equations, we show that DEQGAN can obtain multiple orders of magnitude lower mean squared errors than PINNs that use $L_2$, $L_1$, and Huber loss functions. We also show that DEQGAN achieves solution accuracies that are competitive with popular numerical methods. Finally, we present two methods to improve the robustness of DEQGAN to different hyperparameter settings.

Raphaël Pellegrin, Blake Bullwinkel, Marios Mattheakis et al.

Physics-Informed Neural Networks (PINNs) offer a promising approach to solving differential equations and, more generally, to applying deep learning to problems in the physical sciences. We adopt a recently developed transfer learning approach for PINNs and introduce a multi-head model to efficiently obtain accurate solutions to nonlinear systems of ordinary differential equations with random potentials. In particular, we apply the method to simulate stochastic branched flows, a universal phenomenon in random wave dynamics. Finally, we compare the results achieved by feed forward and GAN-based PINNs on two physically relevant transfer learning tasks and show that our methods provide significant computational speedups in comparison to standard PINNs trained from scratch.

Blake Bullwinkel, Kristen Grabarz, Lily Ke et al.

Differentially private (DP) synthetic data is a promising approach to maximizing the utility of data containing sensitive information. Due to the suppression of underrepresented classes that is often required to achieve privacy, however, it may be in conflict with fairness. We evaluate four DP synthesizers and present empirical results indicating that three of these models frequently degrade fairness outcomes on downstream binary classification tasks. We draw a connection between fairness and the proportion of minority groups present in the generated synthetic data, and find that training synthesizers on data that are pre-processed via a multi-label undersampling method can promote more fair outcomes without degrading accuracy.