Deep Learning for Computational Physics and Game Theory
Abstract
Numerical methods (including computational linear algebra, optimization, numerical methods for ordinary and partial differential equations, etc.) have had a large impact on not only science and engineering but also industry and commerce. This has been facilitatedby increases in the power of computational hardware as well as the development of new numerical approaches and algorithms. Recently, machine and deep learning has risen to become the most popular new algorithm for such advances. Interestingly, many of these machine and deep learning approaches are not entirely new (dating back decades, starting in 1943) and were tried and abandoned over the years; however, modern sophistication in numerical methods and astounding advances in computer architecture have reopened the door onwhat is possible. Unfortunately, some of the rampant success has been achieved with a sloppy understanding of numerical methods (and the underlying mathematics) while leveraging modern day computer hardware to demonstrate impressive results. This has led to approaches that are over-trained or over-fit to cherry-picked data, misleading on their own accuracy (our recent work shows that TensorFlow and PyTorch have unbounded errors on quite simple problems), and quite easily adversarially attacked. Like all other numerical methods, machine and deep learning algorithms can be improved to be more accurate robust; but, this will require a significant increase in mathematical rigor (and less of the look-what-I-can do mentality).The goal of this proposal is to take a more rigorous mathematical approach to selected problems (chosen for their relevance and importance) in computational physics and game theory. Since our lab has studied many computational physics phenomena at length (solids, fluids, solid/fluid coupling, compressible and incompressibleflows, fracture, interfaces, heat flow, phase change, etc.), we have quite a few well-developed different numerical approaches and application areas tochoose from. The key research vehicle here will revolve around understanding how to make such approaches differentiable, understanding where and when they are theoretically not differentiable, and designing numerical techniques to address the lack of differentiability (when/if possible). Our prior work has shown that the roots of a quadratic equation are not differentiable with respect to its coefficients; nonetheless, we successfully addressed this lack of differentiability and designed a robust numerical optimization approach that works robustly for any test case (even when all the coefficients are identically zero). In the same paper, we extended these results to cubic equations, which are of utmost importance when dealing with contact, collision, and interference between various objects in computational physics (and common in real-word scenarios of interest). We have recently started to extend this work to address various problems in game theory involving multi-agent systems wherethe trajectory of each agent can be modeled by a cubic spline and one cares about both proximity (because agents can sometimes interact if close enough) and collisions (which may be sought or avoided). Approved for Public Release.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Nov 09, 2024
- Source ID
- N000142412644
Entities
People
- Ronald Fedkiw
Organizations
- Office of Naval Research
- Stanford University
- United States Navy