Neural network methods for PDEs and Perron-Frobenius operator problems: Neumann series, invariant densities, and test-space adaptivity
Publication Date
July 31, 2026
Creator
Tanakorn Udomworarat
School of Mathematical Sciences, University of Nottingham
Abstract
Deep learning has become a popular area of research due to the high expressivity and scalability of neural networks, which make them powerful tools for solving many types of problems. Alongside these developments, problems related to Perron-Frobenius operators (or transfer operators) have been extensively studied and applied across various fields. Developing neural network methods for these operators would offer significant advantages, particularly in solving complex or high-dimensional problems.
This thesis aims to develop a deep-learning framework for solving problems related to these operators and partial differential equations (PDEs). Specifically, we develop neural network methods to approximate solutions to Neumann series and invariant density problems involving the Perron-Frobenius operator. The core methodologies investigated in this study include physics-informed neural networks (PINNs), variational PINNs (VPINNs), and robust VPINNs (RVPINNs). We further aim to design an adaptive version of the RVPINN framework for PDEs to improve the convergence rate by adaptively refining the test-space mesh.
The first major contribution of this study involves the Neumann series of non-expansive Perron-Frobenius operators. We use PINNs and RVPINNs to approximate solutions in their strong and variational forms, respectively. We provide a priori error estimates for quasi-minimizers of the associated loss functions. We present some numerical results for 1D, 2D, and high-dimensional examples to show the performance of our methods. We also demonstrate the applicability of our methods by approximating interior densities in a two-cavity system.
In the context of invariant density problems, we introduce three loss functions for approximating Perron-Frobenius invariant densities. These loss functions are specifically designed to enforce the constraints from the maximum entropy method (MEM). Furthermore, we derive the analytical gradients when using a two-layer rectified linear unit (ReLU) neural network to improve computational speed. We demonstrate the efficacy of our method through 1D examples, highlighting its ability to approximate invariant densities where traditional discretization techniques might be inefficient.
Finally, we establish theoretical and practical adaptive strategies for the RVPINN framework. We present upper bounds for approximation errors based on the RVPINN loss function and prove the equivalence between these bounds. We then develop adaptive algorithms guided by a Riesz discrepancy indicator and provide their convergence results. Furthermore, we propose a computable refinement indicator and prove that, under the saturation assumption, it serves as a reliable and efficient error estimator for the discrepancy between the discrete and continuous Riesz representatives. We demonstrate the effectiveness of the proposed methodology by solving three problems: a Poisson equation on a square domain with a smooth solution, an elliptic interface problem with a discontinuous coefficient, and a Poisson equation on an L-shaped domain with a singular solution.
Item Type
ethesis
Thesis Type
PhD
Supervisors
School of Mathematical Sciences, University of Nottingham
Richter, Martin
School of Mathematical Sciences, University of Nottingham
Brevis, Ignacio
School of Mathematical Sciences, University of Nottingham
Rojas, Sergio
School of Mathematics, Monash University
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