Neural Superposition Networks

Atiyo Ghosh, Nicolò Toscano, Jongyeong Lee, Hyukgeun Cha, Jun-Ho Lee, Jung Jun Park, Yunjun Choi, Seong-Hyok Sean Kim, Antonio Andrea Gentile

Keywords: differential equations, physics-informed neural networks, scientific machine learning, differentially constrained architecture, principle of superposition

Abstract:

We introduce _Neural Superposition Networks_, a class of physics-constrained neural architectures that exactly satisfy given partial differential equations (PDEs) by construction. In contrast to traditional physics-informed neural networks (PINNs), which enforce PDE constraints via loss regularization, our approach embeds the solution manifold directly into the architecture by expressing the output as a superposition of analytical basis functions that solve the target PDE. This eliminates the need for interior residual loss terms, simplifies training to a single-objective optimization on boundary conditions, and improves numerical stability.
We show that for linear PDEs—including Laplace, heat, and incompressible flow constraints—this architectural bias leads to provably convergent approximations. Using maximum principles and classical convergence theory, we establish uniform boundary-to-interior convergence guarantees. For nonlinear PDEs such as Burgers’ equation, we demonstrate that partial structural constraints can still be enforced via transformations (e.g., Cole–Hopf), yielding improved inductive bias over standard PINNs. The resulting networks combine the expressiveness of deep learning with the convergence guarantees of Galerkin and spectral methods. Our framework offers a theoretically grounded and computationally efficient alternative to residual-based training for PDE-constrained problems..

Primary Area: Machine learning for sciences (e.g. climate, health, life sciences, physics, social sciences)

Submission Number: 15527