# Collaborative Research: Geometric Scientific Machine Learning for PDEs with Tensorial Constraints

> **NSF 01002627DB NSF RESEARCH & RELATED ACTIVIT** · University of Pennsylvania (PA) · $350,000

## Abstract

As artificial intelligence (AI) increasingly accelerates scientific discovery and engineering design, there is a growing need for models that are not only computationally fast but physically reliable. Many current AI approaches rely purely on massive datasets, predicting physical phenomena without incorporating the underlying laws of nature. This purely data-driven approach can lead to predictions that are unstable or physically impossible. This project develops 'physics-preserving' machine learning models that embed geometric and physical constraints directly into the AI's architecture. By ensuring these models obey fundamental physical laws by design, the research yields simulators that operate thousands of times faster than traditional computational methods without sacrificing accuracy. These advancements directly support Federal strategic interests in artificial intelligence and advanced manufacturing by enabling the creation of real-time, highly accurate digital twins for complex systems in aerospace, materials science, and energy. Additionally, the project supports workforce development by training a new generation of scientists, spanning high school, undergraduate, and graduate levels at the critical intersection of computational mathematics and machine learning.


This project will create structure-preserving scientific machine learning (SciML) architectures to learn reduced partial differential equation (PDE) models incorporating constrained tensors. Examples in

## Key facts

- **NSF award ID:** 2608776
- **Awardee organization:** University of Pennsylvania (PA)
- **SAM.gov UEI:** GM1XX56LEP58
- **PI:** Nathaniel A Trask
- **Primary program:** 01002627DB NSF RESEARCH & RELATED ACTIVIT
- **All programs:** Artificial Intelligence (AI), Machine Learning Theory, COMPUTATIONAL SCIENCE & ENGING
- **Estimated total:** $350,000
- **Funds obligated:** $350,000
- **Transaction type:** Standard Grant
- **Period:** 06/15/2026 → 05/31/2029

## Primary source

NSF Award Search: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2608776

## Citation

> US National Science Foundation, Award 2608776, Collaborative Research: Geometric Scientific Machine Learning for PDEs with Tensorial Constraints. Retrieved via AI Analytics 2026-06-08 from https://api.ai-analytics.org/grant/nsf/2608776. Licensed CC0.

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