Scott T. Miller

Francesco Costanzo, Scott T. Miller

A finite element formulation is developed for a poroelastic medium consisting of an incompressible hyperelastic skeleton saturated by an incompressible fluid. The governing equations stem from mixture theory and the application is motivated by the study of interstitial fluid flow in brain tissue. The formulation is based on the adoption of an ALE perspective. We focus on a flow regime in which inertia forces are negligible. The stability and convergence of the formulation is discussed, and numerical results demonstrate agreement with the theory.

Anil Radhakrishnan, John F. Lindner, Scott T. Miller et al.

In contrast to conventional artificial neural networks, which are structurally static, we present two approaches for evolving small networks into larger ones during training. The first method employs an auxiliary weight that directly controls network size, while the second uses a controller-generated mask to modulate neuron participation. Both approaches optimize network size through the same gradient-descent algorithm that updates the network's weights and biases. We evaluate these growing networks on nonlinear regression and classification tasks, where they consistently outperform static networks of equivalent final size. We then explore the hyperparameter space of these networks to find associated scaling relations relative to their static counterparts. Our results suggest that starting small and growing naturally may be preferable to simply starting large, particularly as neural networks continue to grow in size and energy consumption.

Anshul Choudhary, John F. Lindner, Elliott G. Holliday et al.

Conventional neural networks are universal function approximators, but because they are unaware of underlying symmetries or physical laws, they may need impractically many training data to approximate nonlinear dynamics. Recently introduced Hamiltonian neural networks can efficiently learn and forecast dynamical systems that conserve energy, but they require special inputs called canonical coordinates, which may be hard to infer from data. Here we significantly expand the scope of such networks by demonstrating a simple way to train them with any set of generalised coordinates, including easily observable ones.

Scott T. Miller, John F. Lindner, Anshul Choudhary et al.

We detail how incorporating physics into neural network design can significantly improve the learning and forecasting of dynamical systems, even nonlinear systems of many dimensions. A map building perspective elucidates the superiority of Hamiltonian neural networks over conventional neural networks. The results clarify the critical relation between data, dimension, and neural network learning performance.