Andrew S. Gearhart

Alexander New, Benjamin Eng, Andrea C. Timm et al.

In this work, we assess the ability of physics-informed neural networks (PINNs) to solve increasingly-complex coupled ordinary differential equations (ODEs). We focus on a pair of benchmarks: discretized partial differential equations and harmonic oscillators, each of which has a tunable parameter that controls its complexity. Even by varying network architecture and applying a state-of-the-art training method that accounts for "difficult" training regions, we show that PINNs eventually fail to produce correct solutions to these benchmarks as their complexity -- the number of equations and the size of time domain -- increases. We identify several reasons why this may be the case, including insufficient network capacity, poor conditioning of the ODEs, and high local curvature, as measured by the Laplacian of the PINN loss.

Miller Wilt, Jordan K. Matelsky, Andrew S. Gearhart

Federated machine learning is a technique for training a model across multiple devices without exchanging data between them. Because data remains local to each compute node, federated learning is well-suited for use-cases in fields where data is carefully controlled, such as medicine, or in domains with bandwidth constraints. One weakness of this approach is that most federated learning tools rely upon a central server to perform workload delegation and to produce a single shared model. Here, we suggest a flexible framework for decentralizing the federated learning pattern, and provide an open-source, reference implementation compatible with PyTorch.

Susama Agarwala, Benjamin Dees, Andrew Gearhart et al.

We study the deformation of the input space by a trained autoencoder via the Jacobians of the trained weight matrices. In doing so, we prove bounds for the mean squared errors for points in the input space, under assumptions regarding the orthogonality of the eigenvectors. We also show that the trace and the product of the eigenvalues of the Jacobian matrices is a good predictor of the MSE on test points. This is a dataset independent means of testing an autoencoder's ability to generalize on new input. Namely, no knowledge of the dataset on which the network was trained is needed, only the parameters of the trained model.