Physics-Guided Problem Decomposition for Scaling Deep Learning of High-dimensional Eigen-Solvers: The Case of Schrödinger's Equation

02/12/2022
by   Sangeeta Srivastava, et al.
17

Given their ability to effectively learn non-linear mappings and perform fast inference, deep neural networks (NNs) have been proposed as a viable alternative to traditional simulation-driven approaches for solving high-dimensional eigenvalue equations (HDEs), which are the foundation for many scientific applications. Unfortunately, for the learned models in these scientific applications to achieve generalization, a large, diverse, and preferably annotated dataset is typically needed and is computationally expensive to obtain. Furthermore, the learned models tend to be memory- and compute-intensive primarily due to the size of the output layer. While generalization, especially extrapolation, with scarce data has been attempted by imposing physical constraints in the form of physics loss, the problem of model scalability has remained. In this paper, we alleviate the compute bottleneck in the output layer by using physics knowledge to decompose the complex regression task of predicting the high-dimensional eigenvectors into multiple simpler sub-tasks, each of which are learned by a simple "expert" network. We call the resulting architecture of specialized experts Physics-Guided Mixture-of-Experts (PG-MoE). We demonstrate the efficacy of such physics-guided problem decomposition for the case of the Schrödinger's Equation in Quantum Mechanics. Our proposed PG-MoE model predicts the ground-state solution, i.e., the eigenvector that corresponds to the smallest possible eigenvalue. The model is 150x smaller than the network trained to learn the complex task while being competitive in generalization. To improve the generalization of the PG-MoE, we also employ a physics-guided loss function based on variational energy, which by quantum mechanics principles is minimized iff the output is the ground-state solution.

READ FULL TEXT

page 2

page 6

research
11/14/2022

Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computing

Recent breakthroughs in computing power have made it feasible to use mac...
research
04/22/2023

Physics-guided generative adversarial network to learn physical models

This short note describes the concept of guided training of deep neural ...
research
07/02/2020

Learning Neural Networks with Competing Physics Objectives: An Application in Quantum Mechanics

Physics-guided Machine Learning (PGML) is an emerging field of research ...
research
08/18/2023

HyperLoRA for PDEs

Physics-informed neural networks (PINNs) have been widely used to develo...
research
02/23/2023

Q-Flow: Generative Modeling for Differential Equations of Open Quantum Dynamics with Normalizing Flows

Studying the dynamics of open quantum systems holds the potential to ena...
research
07/03/2022

Variational energy based XPINNs for phase field analysis in brittle fracture

Modeling fracture is computationally expensive even in computational sim...
research
03/12/2022

Energy networks for state estimation with random sensors using sparse labels

State estimation is required whenever we deal with high-dimensional dyna...

Please sign up or login with your details

Forgot password? Click here to reset