Fourier Neural Operator

Solutions to many partial differential equations (PDEs) display coexisting smooth global transport and localized sharp features within a single trajectory: shock fronts, thin interfaces, and concentrated high-frequency content sit on top of slowly varying backgrounds. This poses a challenge for neural operators: Fourier-based architectures mix nonlocal interactions efficiently but tend to under-resolve localized non-smooth features, whereas spatially local architectures recover fine detail at the cost of long-range propagation and rollout stability. Existing hybrid operators paper over this tension with a fixed, spatially uniform fusion that forces the same trade-off everywhere. We propose U-HNO, a U-shaped hybrid neural operator whose central design is Sparse-Point Adaptive Routing (SPAR): at every spatial location, a per-pixel hard mask selects whether the global Fourier branch or the local multi-scale Gaussian branch should dominate, and the sparsity ratio is a function of the local contrast of the routing signal, so smooth and shock-aligned regions receive different mixtures of global and local computation. SPAR is embedded in a hierarchical encoder-bottleneck-decoder backbone with skip connections so that the dual branches and the gate operate at every resolution. Training combines pointwise supervision with a finite-difference H^1 gradient term and a band-wise spectral consistency regularizer. Across benchmarks spanning 1D Burgers, Kuramoto-Sivashinsky, KdV, 2D advection, Allen-Cahn, Navier-Stokes, Darcy flow, and 3D transonic compressible Navier-Stokes from PDEBench, U-HNO achieves state-of-the-art rollout accuracy on the majority of tasks in both relative L^2 and H^1 metrics, with the largest gains on problems dominated by sharp localized features. Ablations show that removing any single component substantially degrades rollout error.

Spectral neural operators achieve strong performance for PDE learning, but rely on fixed global bases that limit their ability to represent spatially heterogeneous and multiscale dynamics. We propose Adaptive Basis Learning (ABLE), a framework that learns data-dependent spectral representations instead of relying on predefined bases. ABLE constructs a spatially adaptive Parseval frame via a learned ancillary density, enabling the operator to act in a lifted spectral space while preserving invertibility and maintaining $O(N\log N)$ complexity through FFT-based implementation. This shifts the source of expressivity from spectral coefficients to the representation itself, allowing the model to capture localized structures and non-translation-invariant interactions more efficiently. ABLE integrates seamlessly into existing neural operator architectures as a drop-in replacement for spectral layers. Across a range of benchmarks ABLE improves accuracy over strong baselines, with the largest gains in regimes characterized by sharp gradients and multiscale behavior. Moreover, augmenting existing models (e.g., U-FNO, HPM) with ABLE further enhances their performance, demonstrating its role as a general and complementary spectral refinement. Our results highlight that the data-driven choice of representation, rather than operator complexity alone, is a key bottleneck in neural operator design. By learning the basis itself, ABLE provides a principled and efficient framework for improving spectral methods in PDE learning.

Neural operators provide fast surrogate models for time-dependent partial differential equations, but their standard autoregressive use usually assumes that the instantaneous field $u(t,\cdot)$ is a complete state. This assumption fails for delay equations, distributed-memory systems, and other non-Markovian dynamics: two trajectories may agree at time $t$ and nevertheless have different futures because their histories differ. We introduce the History-Space Fourier Neural Operator (HS-FNO), a neural operator for delay and memory-driven PDEs formulated on the lifted state $u_t(θ,x)=u(t+θ,x)$, $θ\in[-τ,0]$. The key computational step is to decompose one history-state update into a learned predictor for the newly exposed future slice and an exact shift-append transport for the portion of the history window already known from the previous state. This avoids learning deterministic history coordinates, reduces the learned output dimension, and enforces the natural discrete history update. We test HS-FNO on five benchmark families covering delayed reaction--diffusion, spatial epidemiology, nonlocal neural-field dynamics, delayed waves, and distributed-memory closures. Across ten random seeds, HS-FNO attains the lowest aggregate one-step, history-space, and rollout errors among the principal baselines. The largest gain occurs in autoregressive prediction, where aggregate rollout error decreases from $0.241$, $0.188$, and $0.185$ for current-state, lag-stack, and unconstrained history-to-history operators, respectively, to $0.094$. The same model uses fewer parameters than unconstrained history prediction. These results indicate that enforcing the discrete shift structure of history-state evolution is an effective inductive bias for non-Markovian PDE surrogate modeling.

Fourier Neural Operators (FNOs) can greatly accelerate PDE simulation, but they are often used without formal guarantees that they preserve basic physical structure. We show that, once the trained weights and grid are fixed, the spectral convolution in an FNO is a linear map. As a result, the full forward pass is piecewise-linear and can be represented exactly in Z3's linear real arithmetic. We study two encodings. The exact encoding compiles the spectral convolution into a dense matrix multiplication, which is sound for both proofs and counterexamples. The lighter frozen encoding replaces the spectral path with a constant, making it faster but approximate. On 10 small FNO surrogates for 1D advection-diffusion-reaction (85 to 117 parameters, grids 8 to 32), the exact encoding gives 2 sound positivity proofs on linear (ReLU-free) models, 5 sound positivity counterexamples, and 10 sound mass-violation counterexamples; the remaining 3 positivity queries on ReLU models time out. For mass non-increase, Z3 finds worse counterexamples than both gradient-based falsification and Monte Carlo on 7 of 10 models. The frozen encoding scales to grid size 64 with sub-second positivity checks, but it no longer provides certificates for the original FNO. Overall, the results make the soundness--scalability tradeoff explicit and point to what is needed for formal verification of production-scale neural operators.

Neural operators (NOs) are designed to learn maps between infinite-dimensional function spaces. We propose a novel reframing of their use. By introducing an auxiliary base-space, any finite-dimensional function can be viewed as an operator acting by composition on functions of the base-space. Through a range of benchmarks on analytic functions of increasing complexity and dimensionality, we demonstrate that NOs can match or outperform standard multilayer perceptrons and Kolmogorov--Arnold Networks in accuracy while requiring significantly fewer parameters and training time. As a real-world application, we apply a two-dimensional Tensorized Fourier Neural Operator (TFNO) to the nuclear chart, learning a correction to state-of-the-art nuclear mass models as a partially observed residual field. A TFNO ensemble reaches a held-out root-mean-square error of 198.2 keV, placing it among the best recent neural-network approaches while retaining high parameter efficiency and short training times. More broadly, these results introduce NOs as a scalable framework for finite-dimensional function interpolation, from analytic benchmarks to structured scientific data.

Neural operators have emerged as a powerful tool for learning mappings between infinite-dimensional function spaces. However, their approximation properties in Sobolev norms remain poorly quantified, even though these norms control both function values and derivatives and are the natural metrics for PDE well-posedness, stability, and generalization. We develop a functional-analytic framework for operator learning in Sobolev spaces and connect it to the numerical behavior of Fourier Neural Operators (FNOs) on a prototypical PDE. First, for a continuous nonlinear operator $\mathcal{G}: H^{s}(D)\to H^{t}(D')$ with $s > d/2$ and inputs restricted to a compact subset of $H^{s}(D)$, we prove that $\mathcal{G}$ can be uniformly approximated in $H^{t}$-norm by a neural operator with $\mathcal{O}(\varepsilon^{-d/s})$ trainable parameters. This yields an explicit complexity--error relation of the form $\|\mathcal{G}-\mathcal{G}_θ\|_{H^{t}} \lesssim C N^{-s/d}$. We then study the one-dimensional viscous Burgers solution operator $\mathcal{G}: u_{0}\mapsto u(\cdot,1)$ on a bounded $H^{1}$-ball and train FNOs with an $H^{1}$-loss. Across a sweep of model sizes, we obtain test $H^{1}$-errors down to $\mathcal{O}(10^{-7})$ and relative errors of order $10^{-3}$, with predictions accurately matching both solutions and spatial derivatives on held-out data. A log-log plot of Sobolev error versus parameter count exhibits an approximate power law $\|\mathcal{G}-\mathcal{G}_θ\|_{H^{1}} \approx C N^{-α}$ with empirical exponent $α\approx 1.4$, and long-horizon training reveals optimization instabilities in large FNOs, providing quantitative evidence that Sobolev-space approximation theory meaningfully predicts neural-operator scaling behavior.

Fourier Neural Operators are deep learning models that learn mappings between function spaces and can be used to learn and solve partial differential equations (PDEs), in some cases significantly faster than traditional PDE solvers. Within the model are Fourier layers, which apply linear transformations directly to the Fourier modes, with parameters depending on the wave numbers. However, most physical systems are isotropic, with the results being independent of the coordinate system chosen, but the linear transformations do not necessarily respect these symmetries. We propose a modification to the linear transformations that ensures spatial symmetries are respected, called the Isotropic Fourier Neural Operator, which both improves model performance and reduces the number of parameters by up to a factor of 16 in 2D and 96 in 3D.

General-relativistic magnetohydrodynamic (GR-MHD) simulations are essential for studying black hole accretion, relativistic jets, and magnetic reconnection, yet their computational cost severely limits systematic parameter exploration. We investigate neural operator surrogates for two astrophysically relevant simulation scenarios produced by the Black Hole Accretion Code (\texttt{BHAC}). First, a Physics Informed Fourier Neural Operator (PINO) is trained on the special-relativistic resistive MHD (SRRMHD) evolution of the Orszag-Tang vortex over a range of resistivities spanning the Sweet-Parker and fast reconnection regimes. By embedding the governing equations as an additional loss term evaluated at finer temporal resolution than the available data supervision, the model learns dynamics at time steps where no simulation data is provided, enabling recovery of plasmoid formation that a data-only baseline trained on the same sparse snapshots fails to reproduce. To our knowledge, the present work is the first application of a physics informed neural operator to special relativistic resistive MHD, and the first to investigate the capability of such models to resolve plasmoid formation in SRRMHD. In a second line of investigation, an OFormer-style Transformer Neural Operator is trained on the evolution of spine-sheath relativistic jets created with \texttt{BHAC}, in special-relativistic MHD (SRMHD). The model is directly applied on the adaptive mesh, highlighting the need for linear attention due to long sequences. The neural surrogate model is capable of capturing most of the major details, especially in early predictions. To our knowledge, this constitutes the first application of a neural operator directly on a high resolution adaptive mesh refinement grid in the context of MHD simulations.

Multi-scale dynamic wake modeling and prediction are essential for the real-time control and optimization of floating offshore wind turbines (FOWTs). In this study, wakes of FOWTs under coupled surge and pitch motions across a range of Strouhal numbers (St), which can induce wake meandering, are modeled via two novel deep-learning frameworks: physics-informed neural networks (PINNs) and Fourier neural operators (FNOs). The high-fidelity dataset is obtained from large-eddy simulations with the actuator line model (LES-AL). The results demonstrate that the dominant large-scale dynamic structures, such as meandering, can be well modeled by both frameworks; however, FNOs exhibit significant advantages over the PINN model in terms of efficiency (8-fold computational speedup and 40-fold faster convergence), long-term predictive capability, and multi-scale coherent structural fidelity. Furthermore, the wakes predicted by the PINN model exhibit a smoothing effect that limits the resolution of high-frequency coherent structures and underestimates turbulent fluctuations in both the wake center and half-width. Spectral analysis reveals that FNOs resolve the primary meandering frequency (where Stp denotes the frequency induced by the coupled surge and pitch motions), its corresponding higher-order harmonics (2Stp, 3Stp), and the energy cascade. In contrast, the energy cascade in the PINN predictions dissipates more rapidly in the high-frequency regime (St > 1.0). Additionally, the pre-multiplied power spectral density indicates that the energy contained in meandering and the corresponding harmonic frequencies modeled by PINNs is relatively low compared to that in CFD and FNOs. These findings suggest that FNOs are promising for the high-fidelity, real-time modeling of FOWT wakes.

Meta-optics promises compact, high-performance imaging and color routing. However, designing high-performance structures is a high-dimensional optimization problem: mapping a desired optical output back to a physical 3D structure requires solving computationally expensive Maxwell's equations iteratively. Even with adjoint optimization, broadband design can require thousands of Maxwell solves, making industrial-scale optimization slow and costly. To overcome this challenge, we propose the Neural Adjoint Method, a solver-supervised surrogate that predicts 3D adjoint gradient fields from a voxelized permittivity volume using a Fourier Neural Operator (FNO). By learning the dense, per-voxel sensitivity field that drives gradient-based updates, our method can replace per-iteration adjoint solves with fast predictions, greatly reducing the computational cost of full-wave simulations required during iterative refinement. To better preserve sensitivity peaks, we introduce a stage-wise FNO that progressively refines residual errors with increasing emphasis on higher-frequency components. We curate a meta-optics dataset from paired forward/adjoint FDTD simulations and evaluate it across three tasks: spectral sorting (color routers), achromatic focusing (metalenses), and waveguide mode conversion. Our method reduces design time from hours to seconds. These results suggest a practical route toward fast, large-scale volumetric meta-optical design enabled by AI-accelerated scientific computing.

Neural operators (NOs) are powerful deep learning frameworks designed to learn solution operators of partial differential equations. This study evaluates the ability of NOs' to capture the stiff spatio-temporal dynamics of the FitzHugh-Nagumo model. A key contribution of this study is the assessment of the translation invariance using a novel training strategy. Models are trained using an applied current with varying spatial locations and intensities at a fixed time, while the test set presents a challenging out-of-distribution scenario where the current is translated in both time and space. This approach significantly reduces dataset generation costs. We benchmark seven NO architectures: Convolutional Neural Operators (CNOs), Deep Operator Networks (DeepONets), DeepONets with CNN encoders, Proper Orthogonal Decomposition DeepONets, Fourier Neural Operators (FNOs), Tucker Tensorized FNOs, and Local Neural Operators. We evaluated these models based on their accuracy, efficiency, and inference speed. These results demonstrate that CNOs generalize well to translated test dynamics, whereas other architectures do not generalize well. On the training set, all architectures achieve comparable accuracy, with FNOs achieving the highest precision. However, this higher accuracy comes at an elevated computational cost. Meanwhile, DeepONets and their variants exhibit superior training and inference efficiency. These findings highlight the capabilities and limitations of NOs in modeling complex ionic dynamics and provide a comprehensive benchmark for scenarios involving translated dynamics.

This review examines neural operator architectures for learning solution operators of parametric partial differential equations (PDEs), with an emphasis on conceptual clarity and practical implementation. The work analyzes key models, including DeepONet, PCANet, and the Fourier Neural Operator, highlighting their underlying representations, computational structures, and comparative performance. These architectures are demonstrated on three canonical PDE problems: the Poisson equation, a linear elasticity problem, and a hyperelasticity problem. To make the presentation self-contained, key foundational topics are introduced, including finite-dimensional representations of function spaces, singular-value decomposition, and sampling from infinite-dimensional function spaces. Beyond forward modeling, the review discusses the use of neural operators as surrogate models within a Bayesian inverse-problem framework, including prior specification, forward-map approximation, and posterior computation. The performance of the three neural-operator architectures is evaluated on in-distribution samples, out-of-distribution samples, and Bayesian inference tasks. The review also discusses challenges related to prediction accuracy and generalization, outlining emerging strategies such as residual-based error correction and multi-level training. The review concludes by positioning neural operators within broader scientific-computing workflows and by identifying directions for reliable, scalable operator learning.