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Spectral methods for Neural Integral Equations

Emanuele Zappala

TL;DR

The paper introduces a spectral framework for neural integral equations (NIEs) that learns integral operators in the spectral domain using Chebyshev polynomials. By representing solutions with a truncated Chebyshev series $y_N(t)$ and learning the kernel in spectral space, integration reduces to a matrix operation, greatly reducing computational cost while preserving nonlocal modeling capabilities. Under mild regularity assumptions, the authors establish convergence of the projected spectral problem to the true solution and show that the projected operator can be uniformly approximated by neural networks, enabling practical learning of NIEs. Empirical studies on integral-equation data and simulated fMRI demonstrate competitive accuracy, strong interpolation capabilities, and favorable memory/time trade-offs compared to several baselines, with robustness to irregular sampling. The approach has potential applications in neuroscience and other domains featuring nonlocal dynamics.

Abstract

Neural integral equations are deep learning models based on the theory of integral equations, where the model consists of an integral operator and the corresponding equation (of the second kind) which is learned through an optimization procedure. This approach allows to leverage the nonlocal properties of integral operators in machine learning, but it is computationally expensive. In this article, we introduce a framework for neural integral equations based on spectral methods that allows us to learn an operator in the spectral domain, resulting in a cheaper computational cost, as well as in high interpolation accuracy. We study the properties of our methods and show various theoretical guarantees regarding the approximation capabilities of the model, and convergence to solutions of the numerical methods. We provide numerical experiments to demonstrate the practical effectiveness of the resulting model.

Spectral methods for Neural Integral Equations

TL;DR

The paper introduces a spectral framework for neural integral equations (NIEs) that learns integral operators in the spectral domain using Chebyshev polynomials. By representing solutions with a truncated Chebyshev series and learning the kernel in spectral space, integration reduces to a matrix operation, greatly reducing computational cost while preserving nonlocal modeling capabilities. Under mild regularity assumptions, the authors establish convergence of the projected spectral problem to the true solution and show that the projected operator can be uniformly approximated by neural networks, enabling practical learning of NIEs. Empirical studies on integral-equation data and simulated fMRI demonstrate competitive accuracy, strong interpolation capabilities, and favorable memory/time trade-offs compared to several baselines, with robustness to irregular sampling. The approach has potential applications in neuroscience and other domains featuring nonlocal dynamics.

Abstract

Neural integral equations are deep learning models based on the theory of integral equations, where the model consists of an integral operator and the corresponding equation (of the second kind) which is learned through an optimization procedure. This approach allows to leverage the nonlocal properties of integral operators in machine learning, but it is computationally expensive. In this article, we introduce a framework for neural integral equations based on spectral methods that allows us to learn an operator in the spectral domain, resulting in a cheaper computational cost, as well as in high interpolation accuracy. We study the properties of our methods and show various theoretical guarantees regarding the approximation capabilities of the model, and convergence to solutions of the numerical methods. We provide numerical experiments to demonstrate the practical effectiveness of the resulting model.
Paper Structure (8 sections, 4 theorems, 33 equations, 3 figures, 5 tables, 1 algorithm)

This paper contains 8 sections, 4 theorems, 33 equations, 3 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Methods
  3. Theoretical framework and approximation capabilities
  4. Algorithm
  5. Experiments
  6. Integral Equations Dataset
  7. Simulated fMRI Dataset
  8. Conclusions

Key Result

Theorem 3.5

In the hypotheses above, the projected equation admits at least a solution $u_N$ for all $N$ large enough. Moreover, as $N \longrightarrow \infty$, with the same rate as $\|P_Nu - u\| \longrightarrow 0$.

Figures (3)

  • Figure 1: Example prediction for Spectral NIE on fMRI Dataset with $7$ time points used for initialization. Top: Prediction. Middle: Ground truth. Bottom: Error.
  • Figure 2: Example prediction for Spectral NIE on fMRI Dataset with $4$ time points used for initialization. Top: Prediction. Middle: Ground truth. Bottom: Error.
  • Figure 3: Example prediction for Spectral NIE on fMRI Dataset with $7$ time points used for initialization. Top: Prediction. Middle: Ground truth. Bottom: Error.

Theorems & Definitions (15)

  • Definition 3.1
  • Theorem 3.5
  • proof
  • Remark 3.6
  • Remark 3.7
  • Lemma 3.8
  • proof
  • Remark 3.9
  • Theorem 3.10
  • proof
  • ...and 5 more