A functional approach to physics-informed neural networks - Zeiser, Mateus Henrique (2025)
This dissertation investigates the use of energy functionals derived from the variational formulation of partial differential equations (PDEs) as a basis for training Physics-Informed Neural Networks (PINNs). The work begins by revisiting the role of PDEs in modeling physical and biological phenomena, emphasizing the importance of variational principles as a mathematical foundation for obtaining weak solutions and for the development of numerical methods such as the Finite Element Method (FEM). It also reviews essential concepts from Machine Learning, including statistical learning theory, supervised learning, deep neural networks, and the training process based on empirical risk minimization. In this context, the use of automatic differentiation emerges as a key tool for computing gradients efficiently in high-dimensional models. Building on these theoretical elements, we explore the Functional PINN (Fun-PINN), a model that replaces the traditional residual-based training with the minimization of an energy functional directly associated with the PDE, which reduces the order of derivatives required during training, leading to gains in computational efficiency. We also introduce the Self-Adaptive Functional PINN (SA-Fun-PINN), which dynamically adjusts the relative importance of the energy and boundary terms during training. Both models were evaluated against classical PINNs and Self-Adaptive PINNs (SA-PINNs) in numerical experiments designed to assess convergence, accuracy, and computational efficiency. Two test cases were considered: one with a smooth solution, based on Laplace\'s equation, and another with a more oscillatory profile, based on the Poisson equation, allowing performance to be analyzed under increasing levels of complexity. The Fun-PINNs achieved stable training dynamics and offered meaningful reductions in runtime, while the self-adaptive version further improved accuracy with lower computational cost compared to SA-PINNs. In particular, the functional approach demonstrated superior performance in the oscillatory case. However, considering Fun-PINNs as a whole, these experiments do not allow us to claim results superior to those obtained with PINNs; rather, they show that Fun-PINNs are a viable alternative that also yields good results. Overall, this work highlights how classical mathematical concepts such as variational formulations and energy minimization can be effectively integrated with modern machine learning techniques. The proposed functional models combine theoretical consistency with practical efficiency, establishing a promising direction for future research on more complex PDEs, including inverse problems and hybrid approaches that link PINNs with traditional numerical solvers.