Title
From Computational Bottlenecks to Trustworthy AI in Scientific Computing
Abstract
Large-scale scientific simulations are often limited not by the absence of a governing model, but by computational kernels that are expensive and repeatedly evaluated across many grid cells, time steps, and parameter studies. This talk asks how such bottlenecks can be redesigned while preserving the conservation, stability, and physical consistency required for trustworthy scientific computation.
Using subsurface flow as a demanding testbed, I will present three connected themes. First, I will show how locally adaptive sparse-grid surrogates move pressure-temperature flash calculations out of the online compositional-flow simulation. In the reported experiments, the surrogate accelerates the flash-calculation component by approximately 2,000 times, improves the total simulation time by up to 21 times, and reduces stored values by about 90 percent. Second, I will discuss a fully conservative mixed finite-element method for reactive dissolution and wormhole propagation. By treating fluxes as primary unknowns, the method enforces local mass balance and supports stability analysis and a priori error estimates, with numerical results exhibiting nearly first-order convergence. Third, I will describe a physics-informed neural surrogate for flash calculations that incorporates feasible-output constraints, material balance, phase equilibrium, and a physics-based phase decision. This removes the need to generate labeled flash data offline, while revealing an important accuracy challenge near phase-transition boundaries.
I will conclude with a broader research program in which learning proposes computational actions, classical numerical methods verify them, and mathematical analysis provides guarantees for hybrid scientific AI.
Department of Mathematics, Colorado State University
Title
Symmetries in Neural Networks
Abstract
This talk will bring together two topics in machine learning---equivariant neural networks and superposition---under the umbrella of symmetry.
No prior knowledge about neural networks will be assumed.
Although neural networks yield powerful results in many applications, there are drawbacks to their use. For example, there are many fields where the amount of labeled data needed to train neural networks is insufficient. Further, the internal decision-making process of a neural network is often opaque, leading to a desire for more interpretable techniques. These concerns have led some to consider leverage group actions to develop algorithms like equivariant neural networks. We will present theoretical results and numerical testing of a novel neural network structure that enforces partial invariance.
Neural networks have the ability to store more concepts than neurons; this is called superposition. Frames are redundant generalizations of bases
which often have non-trivial symmetry groups. Frame theory has connections to combinatorial design theory, quantum information theory, and more. (In fact,
a recent LLM-driven proof solving the biggest open problem in frame theory took down parts of Hilbert's 12 Problem and the Stark Conjectures in number theory.) Work
done by practitioners investigating trained neural network models showed certain frame-like properties emerge. In this talk, we will some theoretical results
inspired by trained models.
