# PINNIES: An Efficient Physics-Informed Neural Network Framework to Integral Operator Problems

arXiv: https://arxiv.org/abs/2409.01899 · HTML: https://mlsanity.com/paper/2409.01899/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 77 references that are in the map.
- First posted on arXiv 3 Sep 2024.

## Builds on (most cited first, 18 of 18)

- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Optimal Control Via Neural Networks: A Convex Approach](https://mlsanity.com/paper/1805.11835/) (2018)
- [Optimal control of PDEs using physics-informed neural networks](https://mlsanity.com/paper/2111.09880/) (2021)
- [Implicit energy regularization of neural ordinary-differential-equation control](https://mlsanity.com/paper/2103.06525/) (2021)
- [Real-time optimal control via Deep Neural Networks: study on landing problems](https://mlsanity.com/paper/1610.08668/) (2016)
- [Physics-informed neural networks for PDE-constrained optimization and control](https://mlsanity.com/paper/2205.03377/) (2022)
- [AONN: An adjoint-oriented neural network method for all-at-once solutions of parametric optimal control problems](https://mlsanity.com/paper/2302.02076/) (2023)
- [BINN: A deep learning approach for computational mechanics problems based on boundary integral equations](https://mlsanity.com/paper/2301.04480/) (2023)
- [deepFDEnet: A Novel Neural Network Architecture for Solving Fractional Differential Equations](https://mlsanity.com/paper/2309.07684/) (2023)
- [A Comparison of Mesh-Free Differentiable Programming and Data-Driven Strategies for Optimal Control under PDE Constraints](https://mlsanity.com/paper/2310.02286/) (2023)
- [Learning nonlinear integral operators via Recurrent Neural Networks and its application in solving Integro-Differential Equations](https://mlsanity.com/paper/2310.09434/) (2023)
- [Deep Neural Network Solutions for Oscillatory Fredholm Integral Equations](https://mlsanity.com/paper/2401.07003/) (2024)
- [Solving Falkner-Skan type equations via Legendre and Chebyshev Neural Blocks](https://mlsanity.com/paper/2308.03337/) (2023)
- [Solving Elliptic Optimal Control Problems via Neural Networks and Optimality System](https://mlsanity.com/paper/2308.11925/) (2023)
- [Accelerating Fractional PINNs using Operational Matrices of Derivative](https://mlsanity.com/paper/2401.14081/) (2024)
- [UniFIDES: Universal Fractional Integro-Differential Equation Solvers](https://mlsanity.com/paper/2407.01848/) (2024)

## Built on by (most cited first, 3 of 3)

- [A Multimodal Physics-Informed Neural Network Approach for Mean Radiant Temperature Modeling](https://mlsanity.com/paper/2503.08482/) (2025)
- [KANtrol: A Physics-Informed Kolmogorov-Arnold Network Framework for Solving Multi-Dimensional and Fractional Optimal Control Problems](https://mlsanity.com/paper/2409.06649/) (2024)
- [Advanced Physics-Informed Neural Network with Residuals for Solving Complex Integral Equations](https://mlsanity.com/paper/2501.16370/) (2025)

## Abstract (arXiv)

This paper introduces an efficient tensor-vector product technique for the rapid and accurate approximation of integral operators within physics-informed deep learning frameworks. Our approach leverages neural network architectures to evaluate problem dynamics at specific points, while employing Gaussian quadrature formulas to approximate the integral components, even in the presence of infinite domains or singularities. We demonstrate the applicability of this method to both Fredholm and Volterra integral operators, as well as to optimal control problems involving continuous time. Additionally, we outline how this approach can be extended to approximate fractional derivatives and integrals and propose a fast matrix-vector product algorithm for efficiently computing the fractional Caputo derivative. In the numerical section, we conduct comprehensive experiments on forward and inverse problems. For forward problems, we evaluate the performance of our method on over 50 diverse mathematical problems, including multi-dimensional integral equations, systems of integral equations, partial and fractional integro-differential equations, and various optimal control problems in delay, fractional, multi-dimensional, and nonlinear configurations. For inverse problems, we test our approach on several integral equations and fractional integro-differential problems. Finally, we introduce the pinnies Python package to facilitate the implementation and usability of the proposed method.