Scientific machine learning (SciML) has emerged as an impactful discipline for embedding data into science and engineering problems, but often researchers apply "off-the-shelf" ML architectures that fail to exploit substantial available domain expertise and mathematical/physical structure. In the domain of physics-compatible/mimetic PDE discretization, algebraic structure and geometric perspectives have long provided means of designing discrete models which preserve invariances of continuum counterparts (e.g. conservation, involution/gauges, non-trivial null-spaces, maximum principles, entropy stability). We provide an overview of our recent research generalizing these ideas to SciML, designing architectures which incorporate combinatorial Hodge theory and dissipative bracket structures to obtain similar guarantees. This circumvents several pathologies associated with popular schemes incorporating physics weakly via penalty parameters, and may be used to obtain efficient and robust data-driven surrogates and reduced-order models. We also provide a survey of ongoing projects applying these tools to a range of multiscale/multiphysics problems spanning additive manufacturing, mechanics, electromagnetism, shock physics and semiconductor design.

Bio: Dr. Trask is a Senior Member of Technical Staff at in the Center for Computing Research at Sandia National Laboratories. He is the AI/ML lead for the grand challenge LDRD "Beyond Fingerprinting" developing digital twins for additive manufacturing, a recent recipient of the DOE Early Career award and a member of the ASCR MMICCs center PHILMs, where he is developing structure preserving machine learning architectures for use in multiscale modeling of problems spanning a wide range of applications, including shock physics, magnetohydrodynamics, energy storage combustion, material science, and subsurface flow. Prior to converting to permanent staff Nat was the NSF MSPRF postdoctoral fellow at SNL working with Dr. Pavel Bochev. He performed his doctoral studies developing physics compatible meshfree discretizations in the Division of Applied Mathematics at Brown University under the supervision of Profs. Martin Maxey and George Karniadakis, with prior MS + BS degrees in Mechanical Engineering from the University of Massachusetts working in turbulent fuel injection and combustion.