Federal Project Grant Award Summary

The National Science Foundation (NSF) Division of Mathematical Sciences awarded $214,876 to the University of California, Los Angeles (UCLA) under the Mathematical and Physical Sciences program (CFDA 47.049) for a three-year project spanning October 1, 2026 through September 30, 2029. The project develops rigorous theoretical foundations for multi-operator learning—specifically, mathematical frameworks that explain how neural networks can efficiently learn across collections of complex physical systems, particularly nonlinear partial differential equations (PDEs). The research establishes explicit rates for approximation and generalization errors, derives theoretical scalings governing how model size, data, and problem structure affect accuracy, and identifies mechanisms enabling efficient learning in high-dimensional settings.

The deliverables include peer-reviewed theoretical analyses quantifying when a single machine learning model can perform effectively across multiple problems, design principles for neural network architectures tailored to multi-operator learning, and research outputs incorporating structural assumptions to explain sample efficiency gains in high-dimensional prediction settings. The project also contributes to workforce development by training researchers in applied mathematics, applied analysis, and artificial intelligence techniques. These theoretical advances aim to enhance the predictability, interpretability, and robustness of deep learning models for scientific computing while improving resource efficiency—outcomes supporting NSF's broader mission to strengthen the Nation's scientific enterprise through advancement of mathematical and physical sciences.