Translation Invariance of Neural Operators for the FitzHugh-Nagumo Model

Neural operators learn to map initial conditions to the terminal solution of partial differential equations (PDEs), providing a surrogate for the full operator mapping. This enables rapid prediction across different input configurations. While recent neural operator architectures have demonstrated strong performance on diverse PDE tasks, their behavior under structured distribution shifts remains insufficiently understood. To investigate this, we study operator learning in a wave propagation setting governed by a one-dimensional variable-coefficient wave equation, using two representative architectures, the Fourier Neural Operator (FNO) and the Deep Operator Network (DeepONet). To examine their generalization under distribution shifts, we consider structured out-of-distribution (OOD) settings that independently vary input frequency and coefficient smoothness. The results show that under smoothness shifts, both models maintain stable performance, with FNO achieving lower error. In contrast, under frequency shifts, FNO exhibits a sharp increase in error under unseen high-frequency inputs, whereas DeepONet shows milder degradation despite higher overall error. Our analysis reveals that these differences arise from how each architecture represents and responds to variations in frequency structure. Together, these findings highlight a fundamental gap between strong in-distribution performance and generalization under distribution shifts in operator learning, underscoring the role of architectural representation bias in developing more reliable neural operators for physics-based PDE simulations beyond the training distribution.

We introduce the Laplace-Fourier Neural Operator (LFNO), a unified framework for modeling dynamical systems across transient and steady-state regimes by integrating the spectral advantages of Laplace and Fourier Neural Operators. LFNO employs a dual-branch architecture that explicitly decomposes system dynamics into transient and steady-state components. We evaluate LFNO on nine benchmarks, including three ODE systems (Duffing, Lorenz, and Pendulum) and six PDE systems (Euler-Bernoulli beam, Heat, Reaction-diffusion, Brusselator, Burgers, and Navier-Stokes). LFNO significantly outperforms existing operators on ODE systems, where transient dynamics dominate, and consistently surpasses LNO while achieving performance competitive with FNO on PDE benchmarks. Furthermore, LFNO offers improved stability and physical interpretability through its component-wise decomposition. These results demonstrate that LFNO provides a robust and unified approach for learning complex dynamical systems across multiple temporal scales.

Neural operators have emerged as a powerful, discretization-invariant framework for solving partial differential equations (PDEs). Although established approaches like the Deep Operator Network (DeepONet) have successfully achieved universal approximation for operators, and architectures such as Fourier Neural Operators (FNOs) have shown algebraic convergence rates, a precise theoretical connection between the continuous theory and its discrete numerical implementation remains a challenge. Specifically, the relationship between the continuous formulation and the discrete numerical stability has yet to be fully explored. In this paper, we address this gap by establishing theoretical guarantees for the discretization error and stability of neural operator approximation schemes. We prove analytical bounds that link solution regularity to input discretization, providing a formal quantification of neural operator accuracy under real-world numerical constraints. We derive these bounds to the specific cases of State Space Model-based Neural Operators (SS-NOs) and FNOs, thus providing a new discretization error theorem for these models. Additionally, through an input-to-state stability (ISS) analysis, we formally assess the impact of discretization on the stability of SS-NOs results obtained in the continuous domain. Our empirical experiments on 1D and 2D benchmarks validate our theoretical bounds and show the robustness of SS-NOs under varying resolutions.