Operator Learning Based on Geometric Classical Field Theory and Infinite Dimensional Data Science
Operator Learning Based on Geometric Classical Field Theory and Infinite Dimensional Data Science

Research Overview

Overview

Since around 2020, the application of machine learning to scientific computing has been developed rapidly, giving rise to a new research field called Scientific Machine Learning (SciML). In this research project, we focus on one of the methods in SciML, specifically the approach known as operator learning. Operator learning involves using deep learning to learn the solution operator —that is, something analogous to an analytic solution formula— for partial differential equations used in physical simulations. By doing so, a significant acceleration of physical simulations is anticipated.

On the other hand, machine learning techniques often raise concerns about reliability, and the development of reliable methods is needed for operator learning as well. However, because operator learning is a completely different approach from traditional scientific computing methods, its theoretical foundations must be built entirely anew. In particular, operator learning involves learning nonlinear operators, i.e. mappings between infinite-dimensional spaces. Therefore, to establish methods and theory for physical simulation via operator learning, we need to extend various techniques from data science to infinite dimensions and integrate them with the classical field theory (a dynamical theory in infinite dimensions). In this project, we aim to construct such a theory in practice while simultaneously developing operator learning algorithms that preserve properties such as physical laws and passivity.

Operator Learning

Operator learning, applied to the initial/boundary value problems of partial differential equations, accelerates physical simulation by learning the solution operator. The solution operator is, intuitively, a function that maps the simulation's input data (such as initial and/or boundary conditions) to the equation's solution (the simulation result). In other words, operator learning attempts to construct a machine learning model that, when given parameters describing a simulation setup, can immediately output the simulation results.

One of the fundamental questions in the theory of partial differential equations is their well-posedness — namely, the theoretical proof of the existence and uniqueness, and continuous dependence of the solution on initial or boundary data. This guarantees the existence of a continuous function that maps the initial and boundary values to the solution. Meanwhile, it is well known that neural networks can approximate a variety of functions, including continuous functions. Therefore, one can expect that the function providing the solution (whose existence is guaranteed by the well-posedness) can also be approximated by neural networks, and indeed this has been mathematically proven in certain cases.

Learning a solution operator is, intuitively, equivalent to learning a formula for the solution. If this is achieved, it eliminates the need to perform simulations in the traditional manner. Consequently, performing large-scale physical simulations in real time is becoming a reality.

Scientific Machine Learning

The fusion of machine learning and scientific computing is called Scientific Machine Learning (SciML). It is similar to concepts like "AI for Science," but at present, SciML mainly focuses on scientific computing and physical modeling. More specifically, it encompasses research on methods for solving differential equations (which are solved in physical simulations) more efficiently, and methods for deriving equations that describe physical phenomena.

Representative methods include:

  • Approaches for physical modeling, such as neural ordinary differential equations and Hamiltonian neural networks.
  • Approaches for physical simulation, such as Physics-Informed Neural Networks (PINNs) and operator learning.

Just as large-scale language models based on deep learning have had a major societal impact, these SciML methods have the potential to drastically change how we conduct physical simulations. Therefore, this project aims to develop machine learning-based methods for physical simulation and modeling, while simultaneously building the theoretical foundations needed to ensure their reliability.