Publications – Mauro Maggioni's personal home page

2024

Loeffler, Shane E.; Ahmad, Zan; Ali, Syed Yusuf; Yamamoto, Carolyna; Popescu, Dan M.; Yee, Alana; Lal, Yash; Trayanova, Natalia; Maggioni, Mauro

Graph Fourier Neural Kernels (G-FuNK): Learning Solutions of Nonlinear Diffusive Parametric PDEs on Multiple Domains Conference

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}

}

2020

Okada, David Jason Miller; Jonathan Chrispin; Adityo Prakosa; Natalia Trayanova; Steven Jones; Mauro Maggioni; Katherine Wu R ; C David R.

Substrate Spatial Complexity Analysis for the Prediction of Ventricular Arrhythmias in Patients with Ischemic Cardiomyopathy Journal Article

In: Circulation: Arrhythmia and Electrophysiology, 2020.

Links | BibTeX | Tags: imaging, Laplacian eigenfunctions, medical imaging

2010

Jones, Peter W; Maggioni, Mauro; Schul, Raanan

Universal local manifold parametrizations via heat kernels and eigenfunctions of the Laplacian Journal Article

In: Ann. Acad. Scient. Fen., vol. 35, pp. 1–44, 2010, (http://arxiv.org/abs/0709.1975).

BibTeX | Tags: diffusion geometry, heat kernels, Laplacian eigenfunctions, Manifold Learning, multiscale analysis, random walks, spectral graph theory

2008

Coifman, Ronald R; Maggioni, Mauro

Geometry Analysis and Signal Processing on Digital Data, Emergent Structures, and Knowledge Building Miscellaneous

SIAM News, 2008.

BibTeX | Tags: diffusion geometry, heat kernels, Laplacian eigenfunctions, Manifold Learning, multiscale analysis, random walks, spectral graph theory

Maggioni, Mauro; Mhaskar, Hrushikesh

Diffusion polynomial frames on metric measure spaces Journal Article

In: ACHA, vol. 3, pp. 329–353, 2008.

BibTeX | Tags: approximation theory, diffusion geometry, heat kernels, Laplacian eigenfunctions, multiscale analysis

Jones, Peter W; Maggioni, Mauro; Schul, Raanan

Manifold parametrizations by eigenfunctions of the Laplacian and heat kernels Journal Article

In: Proc. Nat. Acad. Sci., vol. 105, no. 6, pp. 1803–1808, 2008.

BibTeX | Tags: diffusion geometry, heat kernels, Laplacian eigenfunctions, Manifold Learning, multiscale analysis, random walks, spectral graph theory

Coifman, Ronald R; Kevrekidis, Ioannis G; Lafon, Stephane; Maggioni, Mauro; Nadler, Boaz

Diffusion Maps, reduction coordinates and low dimensional representation of stochastic systems Journal Article

In: SIAM J.M.M.S., vol. 7, no. 2, pp. 842–864, 2008.

BibTeX | Tags: diffusion geometry, dynamical systems, Laplacian eigenfunctions, Machine learning, model reduction, stochastic systems

2007

Mahadevan, Sridhar; Maggioni, Mauro

Proto-value Functions: A Spectral Framework for Solving Markov Decision Processes Journal Article

In: JMLR, vol. 8, pp. 2169–2231, 2007.

BibTeX | Tags: diffusion geometry, Laplacian eigenfunctions, Machine learning, Manifold Learning, random walks, reinforcement learning, representation learning, spectral graph theory

2006

Coifman, Ronald R; Lafon, Stephane; Maggioni, Mauro; Keller, Y; Szlam, A D; Warner, F J; Zucker, S W

Geometries of sensor outputs, inference, and information processing Proceedings Article

In: Athale, John Zolper; Eds. C Intelligent Integrated Microsystems; Ravindra A. (Ed.): Proc. SPIE, pp. 623209, 2006.

BibTeX | Tags: diffusion geometry, Laplacian eigenfunctions, Machine learning, Manifold Learning, random walks, spectral graph theory, stochastic systems

Maggioni, Mauro; Mahadevan, Sridhar

Fast Direct Policy Evaluation using Multiscale Analysis of Markov Diffusion Processes Proceedings Article

In: ICML 2006, pp. 601–608, 2006.

BibTeX | Tags: diffusion geometry, Laplacian eigenfunctions, Machine learning, Manifold Learning, random walks, reinforcement learning, representation learning, spectral graph theory

Mahadevan, Sridhar; Ferguson, Kim; Osentoski, Sarah; Maggioni, Mauro

Simultaneous Learning of Representation and Control In Continuous Domains Proceedings Article

In: AAAI, AAAI Press, 2006.

BibTeX | Tags: diffusion geometry, Laplacian eigenfunctions, Machine learning, Manifold Learning, random walks, reinforcement learning, representation learning, spectral graph theory

2005

Coifman, Ronald R; Maggioni, Mauro; Zucker, Steven W; Kevrekidis, Ioannis G

Geometric diffusions for the analysis of data from sensor networks Journal Article

In: Curr Opin Neurobiol, vol. 15, no. 5, pp. 576–84, 2005.

BibTeX | Tags: diffusion geometry, Laplacian eigenfunctions, Machine learning, Manifold Learning, random walks, spectral graph theory, stochastic systems

Mahadevan, Sridhar; Maggioni, Mauro

Value Function Approximation with Diffusion Wavelets and Laplacian Eigenfunctions Proceedings Article

In: University of Massachusetts, Department of Computer Science Technical Report TR-2005-38; Proc. NIPS 2005, 2005.

BibTeX | Tags: diffusion geometry, diffusion wavelets, Laplacian eigenfunctions, Machine learning, Manifold Learning, random walks, reinforcement learning, representation learning, spectral graph theory

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.