Research – Computational Physics Group

Scientific Machine Learning

SciML research in the group ranges over several topical areas.

Inference of partial differential equations

Over the past few years we have developed methods to identify partial differential equations from spatiotemporal data. These approaches have been applied to infer PDEs governing the dynamics of cancer cells, pattern formation in materials physics, behavior of soft materials and even to epidemiological data.

A wound healing assay on MDA-MB-231 cells (left) and predictions of an inferred partial differential equation model (right).

Optimal transport and learning

The theory of optimal transport describes the transformation of densities of all kinds: mass, chemical signaling, agents, and more. They may all be interpreted as probability densities, and are subject to a class of PDEs describing the evolution of the densities following gradient descent under a potential. Fokker-Planck PDEs are one example that have wide-ranging applications: from mass and charge transport in biology and materials physics, through chemical signaling to social dynamics. They also lead to connections with learning frameworks–an aspect of special interest to the group.

The Fokker-Planck equation describes transport of densities, and has connections to reinforcement learning.

Machine learning solvers of partial differential equations

While the highest-fidelity solutions of PDEs are provided by discretization-based solvers, trained machine learning methods have advantages of speed for high-throughput applications such as design and optimization at some loss of accuracy. Of particular interest to us is to develop ML solvers that are generalizable across boundary value problems, domains and parameters.

(a) The learnable Bayesian neural network generates a solution that is subject to PDE constraints in weak form by the fixed kernels in (b). The pixelated image is mapped onto finite element nodes ( c) and subject to boundary conditions and assembly in (d).

Foundation models for physics

This is a new initiative for the group, but one with many aspects to it. We are aiming at mathematics-simulation-physics-language foundation models that combine the ability to solve PDEs with inference to explain emergent physics. Watch this space…

Computational Physics

We have developed computational approaches to solve multiphysis problems in biology and materials. Now some of these computational physics frameworks are coupled to the SciML work described above. Some examples follow.

Pattern formation in biology

Nonlinear reaction-diffusion and phase field equations describe the evolution of biological systems, including interacting cell populations. These systems form a diversity of patterns driven by predator-prey and phase transformation dynamics. They have relevance ranging from development of organisms and organs, to the dynamics of migration and signaling of cancer cells.

Nonlinear reaction-diffusion of a two-species cell population.

Scale-bridging in materials physics

We have developed Integrable Deep Neural Networks and active learning algorithms to bridge scale from first-principles statistical mechanics to continuum phase field computations with applications to battery materials.

ML-aided bridging of scales in materials physics.

Coupled physics in battery materials

The group has developed an array of computational models for the coupled electro-chemo-thermo-mechanics of battery materials. Of interest to us have been the formation of morphology and mechanically driven failure of solid state batteries.