Collaborative Research: Unified Neural Operator (UNO): Towards Trustworthy Operator Learning for Scientific Applications

About This Grant

Modern scientific challenges—from predicting complex fluid flows to modeling plasma behavior in fusion reactors—demand computationally efficient, trustworthy surrogates that can rival traditional numerical solvers while harnessing the power of artificial intelligence. Scientific machine learning (SciML), identified as a core technology for AI, offers immense potential for surrogate modeling in both data‑rich and data‑scarce situations; of particular interest is the field of operator learning. However, current operator learning frameworks lack unified theoretical foundations, robustness guarantees, and scalable training methods, which limit their adoption in high-stakes applications. The Unified Neural Operator (UNO) considered in this project will fill this gap by embedding all operator learning techniques into a unifying framework, marrying the mathematical rigor of traditional methods with the expressivity of modern AI. By delivering certifiable, interpretable AI‑driven surrogates, UNO advances Presidential priorities in artificial intelligence and nuclear energy—supporting both next‑generation AI capabilities and efficient modeling of magnetohydrodynamic systems critical for fusion energy—while fulfilling NSF’s mission “to promote the progress of science; to advance the national health, prosperity, and welfare; and to secure the national defense” Within SciML, operator learning has shown tremendous potential as a powerful tool for creating surrogate models, leading to a bevy of deep machine learning (ML)-based operator learning techniques known as “neural operators”. However, poorly-understood robustness characteristics, lack of explainability and interpretability, and the sheer variety of such approaches make it challenging for practitioners to choose the appropriate methods for different tasks, especially in the context of scientific applications. This project tackles these urgent challenges through the inception of a new computational framework: the Unified Neural Operator (UNO). UNO distills neural operators down into three essential components: an input encoder, a set of basis functions for the output space, and a projection operator. The work will (1) provide a mathematical formalism that both encompasses existing neural operators and allows us to generate novel architectures that target specific tasks and problems; (2) provide algorithms for scalable and adaptive training and inference, allowing UNO to adapt to local solution features and to tackle high-dimensional data efficiently in data-rich regimes; (3) provide a robust theoretical foundation in the form of universal approximation theorems, error estimates, and a guiding theoretical framework for robust sampling and adaptivity. The UNO framework also allows for automatic and natural uncertainty quantification capabilities of existing and new neural operators. In data-poor situations, the UNO framework preserves accuracy by analytically preserving physics, thereby making it well-suited to both in situ and ex situ surrogate modeling in scientific applications. The challenging applications targeted by this project include turbulent, multiscale, and multiphysics fluid flow problems. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.