Upcoming Event: Oden Institute Seminar

Generality Meets Precision: Teaching Machine Learning to Love Numerics

Jerry W Liu, Stanford University

Abstract

Scientific machine learning (SciML) promises to bring the generality of foundation models to scientific/engineering domains, but its potential is tempered by fundamental numerical challenges. We discuss two recent works that illustrate both sides of this story. The first explores in-context operator learning, where ideas from large language models are adapted to predict solution operators for differential equations directly from examples, without retraining. We motivate this approach through connections to spectral methods and show encouraging initial results, but also find that achieving high-precision solutions remains a key obstacle. The second investigates this precision barrier in the setting of physics-informed neural networks (PINNs), where standard ML architectures and optimizers saturate well short of double precision. To address this, we introduce the Barycentric Weight Layer (BWLer), a spectral-inspired architecture that restores near machine-precision accuracy and makes visible the classical precision-conditioning tradeoff. Together, these works underscore both the promise and the pitfalls of SciML, and highlight how progress will require numerics-informed architecture and algorithm design.

Biography

erry Liu is a 4th-year Ph.D. candidate in the Institute for Computational & Mathematical Engineering at Stanford University, advised by Professor Chris Ré, and supported by the Department of Energy Computational Science Graduate Fellowship. His research focuses on scientific machine learning, particularly the numerical limitations of modern ML architectures and the development of more principled methods for high-precision computation. Prior to Stanford, he completed undergraduate degrees in Mathematics and Computer Science at Duke University, where he was advised by Cynthia Rudin.