Efficient PDE Solutions using Hartley Neural Operators in Physics-Informed Networks: Potentials and Limitations
Keywords: Hartley Neural Operators (HNO), Physics-Informed Neural Operator Networks (PINOs), Discrete Hartley Transform, Machine Learning, Partial Differential Equations (PDEs)
TL;DR: Hartley Neural Operators in PINOs offer efficient solutions for specific PDEs with improved run-time and adherence to physical laws, but exhibit limitations under complex boundary conditions; a combined approach with Fourier Operators may be optimal.
Abstract: In this work, we introduce novel differentiable architectures for solving partial differential equations (PDEs) using the well-known Discrete Hartley Transform. We incorporate Hartley Neural Operators (HNO) into Physics-Informed Neural Operator Networks (PINOs). Our analysis concentrates on two pivotal PDEs: 1. the one-dimensional diffusion equation, which holds significance not only in machine learning but also across a broad spectrum of physical sciences and engineering disciplines; and 2. the one-dimensional Thermodynamic Energy Equation that is commonly used in weather data analysis. Our implementation of HNO that employs real-valued linear transforms into the PINO architecture results in significant run-time improvements. We show that reconstruction loss is lower than other recently introduced operators that may be used for the above PDEs. Importantly, we find that HNOs naturally satisfy the governing physical laws and equations specific to the PDEs under consideration. However, our empirical observations suggest that the benefits of HNO diminish in certain scenarios where the underlying physical conditions at the boundary are less tractable and involve complex numbers. As an example of a potential failure mode we illustrate that in the case of the one-dimensional Burger's equation, traditional Fourier Neural Operators outperform their Hartley counterparts. Our results indicate that a combination of neural operators including Fourier and Hartley transforms may be better to effectively address the specific type, and/or context of the physical problem at hand.
Primary Area: general machine learning (i.e., none of the above)
Code Of Ethics: I acknowledge that I and all co-authors of this work have read and commit to adhering to the ICLR Code of Ethics.
Submission Guidelines: I certify that this submission complies with the submission instructions as described on https://iclr.cc/Conferences/2024/AuthorGuide.
Anonymous Url: I certify that there is no URL (e.g., github page) that could be used to find authors' identity.
No Acknowledgement Section: I certify that there is no acknowledgement section in this submission for double blind review.
Submission Number: 6854
Loading