2024
Loeffler, Shane E.; Ahmad, Zan; Ali, Syed Yusuf; Yamamoto, Carolyna; Popescu, Dan M.; Yee, Alana; Lal, Yash; Trayanova, Natalia; Maggioni, Mauro
2024.
Abstract | Links | BibTeX | Tags: digital twins, Laplacian eigenfunctions, neural networks, PDEs, precision medicine
@conference{nokey,
title = {Graph Fourier Neural Kernels (G-FuNK): Learning Solutions of Nonlinear Diffusive Parametric PDEs on Multiple Domains},
author = {Shane E. Loeffler and Zan Ahmad and Syed Yusuf Ali and Carolyna Yamamoto and Dan M. Popescu and Alana Yee and Yash Lal and Natalia Trayanova and Mauro Maggioni},
url = {https://doi.org/10.48550/arXiv.2410.04655},
year = {2024},
date = {2024-10-09},
urldate = {2024-10-09},
abstract = {Predicting time-dependent dynamics of complex systems governed by non-linear partial differential equations (PDEs) with varying parameters and domains is a challenging task motivated by applications across various fields. We introduce a novel family of neural operators based on our Graph Fourier Neural Kernels, designed to learn solution generators for nonlinear PDEs in which the highest-order term is diffusive, across multiple domains and parameters. G-FuNK combines components that are parameter- and domain-adapted with others that are not. The domain-adapted components are constructed using a weighted graph on the discretized domain, where the graph Laplacian approximates the highest-order diffusive term, ensuring boundary condition compliance and capturing the parameter and domain-specific behavior. Meanwhile, the learned components transfer across domains and parameters using our variant Fourier Neural Operators. This approach naturally embeds geometric and directional information, improving generalization to new test domains without need for retraining the network. To handle temporal dynamics, our method incorporates an integrated ODE solver to predict the evolution of the system. Experiments show G-FuNK's capability to accurately approximate heat, reaction diffusion, and cardiac electrophysiology equations across various geometries and anisotropic diffusivity fields. G-FuNK achieves low relative errors on unseen domains and fiber fields, significantly accelerating predictions compared to traditional finite-element solvers.},
keywords = {digital twins, Laplacian eigenfunctions, neural networks, PDEs, precision medicine},
pubstate = {published},
tppubtype = {conference}
}
Predicting time-dependent dynamics of complex systems governed by non-linear partial differential equations (PDEs) with varying parameters and domains is a challenging task motivated by applications across various fields. We introduce a novel family of neural operators based on our Graph Fourier Neural Kernels, designed to learn solution generators for nonlinear PDEs in which the highest-order term is diffusive, across multiple domains and parameters. G-FuNK combines components that are parameter- and domain-adapted with others that are not. The domain-adapted components are constructed using a weighted graph on the discretized domain, where the graph Laplacian approximates the highest-order diffusive term, ensuring boundary condition compliance and capturing the parameter and domain-specific behavior. Meanwhile, the learned components transfer across domains and parameters using our variant Fourier Neural Operators. This approach naturally embeds geometric and directional information, improving generalization to new test domains without need for retraining the network. To handle temporal dynamics, our method incorporates an integrated ODE solver to predict the evolution of the system. Experiments show G-FuNK's capability to accurately approximate heat, reaction diffusion, and cardiac electrophysiology equations across various geometries and anisotropic diffusivity fields. G-FuNK achieves low relative errors on unseen domains and fiber fields, significantly accelerating predictions compared to traditional finite-element solvers.
Yin, Minglang; Charon, Nicolas; Brody, Ryan; Lu, Lu; Trayanova, Natalia; Maggioni, Mauro
A scalable framework for learning the geometry-dependent solution operators of partial differential equations Journal Article
In: Nature Computational Science, 2024.
Abstract | Links | BibTeX | Tags: digital twins, Machine learning, model reduction, neural networks, PDEs, precision medicine
@article{DIMON2024,
title = {A scalable framework for learning the geometry-dependent solution operators of partial differential equations},
author = {Minglang Yin and Nicolas Charon and Ryan Brody and Lu Lu and Natalia Trayanova and Mauro Maggioni},
url = {https://arxiv.org/pdf/2402.07250.pdf
https://www.nature.com/articles/s43588-024-00732-2
https://rdcu.be/d2UPp
https://github.com/MinglangYin/DIMON},
doi = {10.1038/s43588-024-00732-2},
year = {2024},
date = {2024-02-13},
urldate = {2024-02-13},
journal = {Nature Computational Science},
abstract = {The solution of a PDE over varying initial/boundary conditions on multiple domains is needed in a wide variety of applications, but it is computationally expensive if the solution is computed de novo whenever the initial/boundary conditions of the domain change. We introduce a general operator learning framework, called DIffeomorphic Mapping Operator learNing (DIMON) to learn approximate PDE solutions over a family of domains ${Omega_{theta}}_theta$, that learns the map from initial/boundary conditions and domain $Omega_theta$ to the solution of the PDE, or to specified functionals thereof. DIMON is based on transporting a given problem (initial/boundary conditions and domain $Omega_theta$) to a problem on a reference domain $Omega_0$, where training data from multiple problems is used to learn the map to the solution on $Omega_0$, which is then re-mapped to the original domain $Omega_theta$. We consider several problems to demonstrate the performance of the framework in learning both static and time-dependent PDEs on non-rigid geometries; these include solving the Laplace equation, reaction-diffusion equations, and a multiscale PDE that characterizes the electrical propagation on the left ventricle. This work paves the way toward the fast prediction of PDE solutions on a family of domains and the application of neural operators in engineering and precision medicine.},
keywords = {digital twins, Machine learning, model reduction, neural networks, PDEs, precision medicine},
pubstate = {published},
tppubtype = {article}
}
The solution of a PDE over varying initial/boundary conditions on multiple domains is needed in a wide variety of applications, but it is computationally expensive if the solution is computed de novo whenever the initial/boundary conditions of the domain change. We introduce a general operator learning framework, called DIffeomorphic Mapping Operator learNing (DIMON) to learn approximate PDE solutions over a family of domains ${Omega_{theta}}_theta$, that learns the map from initial/boundary conditions and domain $Omega_theta$ to the solution of the PDE, or to specified functionals thereof. DIMON is based on transporting a given problem (initial/boundary conditions and domain $Omega_theta$) to a problem on a reference domain $Omega_0$, where training data from multiple problems is used to learn the map to the solution on $Omega_0$, which is then re-mapped to the original domain $Omega_theta$. We consider several problems to demonstrate the performance of the framework in learning both static and time-dependent PDEs on non-rigid geometries; these include solving the Laplace equation, reaction-diffusion equations, and a multiscale PDE that characterizes the electrical propagation on the left ventricle. This work paves the way toward the fast prediction of PDE solutions on a family of domains and the application of neural operators in engineering and precision medicine.
- Lectures at Summer School at Peking University, July 2017.
- PCMI Lectures, Summer 2016: Lecture 1, Lecture 2, Problems/discussion points
- Google Scholar
- Papers on the ArXiv
- Papers on MathsciNet
- Tutorials on diffusion geometry and multiscale analysis on graphs at the MRA Internet Program at IPAM: Part I and Part II.
- Diffusion Geometries, Diffusion Wavelets and Harmonic Analysis of large data sets, IPAM, Multiscale Geometric Analysis Program, Fall 2004.
- Diffusion Geometries, global and multiscale, IPAM, 2005.