PINNIES: An Efficient Physics-Informed Neural Network Framework to Integral Operator Problems
This work addresses computational bottlenecks in physics-informed machine learning for integral operator problems, offering a practical tool for researchers in applied mathematics and engineering, though it is incremental as it builds on existing frameworks with efficiency improvements.
The paper tackles the challenge of efficiently approximating integral operators in physics-informed neural networks by introducing a tensor-vector product technique that uses Gaussian quadrature for integral components, even with infinite domains or singularities, and demonstrates its applicability to over 50 diverse mathematical problems, including forward and inverse problems in integral equations and optimal control.
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.