Neural Operators

Neural operators have emerged as a powerful data-driven approach for solving time-dependent PDEs. Among recent advances, memory-augmented neural operators explicitly incorporate past states and have achieved remarkable performance under low-resolution observation settings. However, existing approaches apply a fixed memory weight regardless of observation conditions, such as resolution or physical parameters, limiting their adaptability. Our preliminary experiments reveal that optimal memory weight varies with resolution and viscosity, implying that a fixed memory weight cannot simultaneously optimize performance across diverse settings. We propose AMGFNO, which dynamically modulates memory weight through a learnable gate. On the Kuramoto-Sivashinsky and Burgers' equations, AMGFNO achieves 55-79% nRMSE reduction over at low resolution, with the learned gate value automatically decreasing from $\bar{g} \approx 0.7$ to near-zero as resolution increases.

Neural operators provide a powerful framework for learning solution mappings of partial differential equations directly in function space. However, many existing architectures still struggle to represent nonlinear time-dependent systems that involve multi-scale structures, long-range interactions, and stable long-time evolution. In this work, we introduce the Hierarchical Adaptive Multi-scale Neural Operator (HAMNO), a neural-operator architecture that combines local convolutional representations, global spectral operators, and hierarchical encoder-decoder processing. The central component of HAMNO is a data-dependent gating mechanism that adaptively balances local and global information at each spatial location, allowing the model to resolve fine-scale features while preserving long-range dependencies. We further develop a physics-informed extension, PI-HAMNO, based on a multi-objective loss strategy that combines data fitting with strong- and weak-form physics constraints. The strong-form term penalizes the domain-integrated squared PDE residual in physical coordinates, while the weak-form term is constructed by multiplying the governing residual by finite-element test functions and evaluating the resulting element integrals using centroid-based tetrahedral quadrature. The framework is evaluated on non-periodic Allen-Cahn (AC), Cahn-Hilliard (CH), and Swift-Hohenberg (SH) equations defined on cubic domains. Across long-horizon rollout, data-limited training, out-of-distribution initial-condition shifts, and random-seed variations, HAMNO improves predictive accuracy over standard neural-operator baselines, while PI-HAMNO further enhances stability, physical consistency, and data efficiency. The implementation is publicly available at https://github.com/MBamdad/HAMNO .

Fourier neural operators (FNOs) are effective and efficient surrogates for approximating solutions of PDEs and generalize across discretizations. However, owing to the reliance on frequency truncation to maintain learning efficiency of FNOs, empirical studies suggest that FNOs exhibit spectral bias toward low-frequency information, which may hinder the learning capability especially for certain PDEs with strong high-frequency oscillations. To address this limitation, we propose SirenFNO, a novel framework that leverages sinusoidal representation networks (SIRENs) to learn implicit neural representations and performs mode-wise kernel parameterization. Our SIREN parameterization learns a full-grid spectrum with a constant and discretization-independent parameter count, thereby eliminating the need for frequency truncation. We further extend SirenFNO with functional tensor decompositions to enhance parameter and learning efficiency. Empirical results show that our SirenFNO consistently outperforms FNO with approximately $4$ to $15$ times parameter reductions with preserved discretization invariance, and our functional decomposition variants obtain performance improvements with a maximum of $73$ times fewer parameters across multiple PDE benchmarks.

We introduce Topological Neural Operators (TNOs), a principled framework for operator learning on cell complexes that lifts neural operators (NOs) from functions on points and/or edges to topological domains. TNOs represent data as features defined on cells of varying dimension and model their interactions through Discrete Exterior Calculus, enabling explicit cross-dimensional coupling via gradient-, curl-, and divergence-type operators. The key design principle is to decouple where information flows, as governed by fixed topological operators, from how it is transformed (which is learned), yielding models that respect the geometric support of physical quantities and expose conservation and compatibility structure. We further propose Hierarchical TNOs (HTNOs), which incorporate learned coarse complexes to propagate long-range and topology-dependent information. Our framework subsumes existing NOs as a special case, providing a unified perspective on operator learning across discretizations. Across a range of PDE benchmarks, including irregular-geometry flow problems, TNOs and HTNOs improve accuracy; controlled studies further isolate the benefits of native higher-rank and topological structure. Project page: https://circle-group.github.io/research/TNO

Aerodynamic simulation is a key component of engineering shape design, where core quantities such as the surface pressure coefficient strongly depend on flow dynamics near solid boundaries. Neural operators provide an efficient alternative to expensive Computational Fluid Dynamics (CFD) solvers. However, conventional methods treat the boundary region isotropically, failing to account for the distinct physical behaviors along the boundaries. In reality, the aerodynamic process exhibits anisotropy: along the tangential direction, flow propagates along the wall; along the normal direction, physical quantities are constrained by the wall. To explicitly model the distinct physical behaviors, we propose GeoABC, a geometry-conditioned anisotropic boundary correction framework. GeoABC leverages the boundary geometries to introduce direction-aware boundary correction into the intermediate representations of neural operators, transforming boundary geometry from static input features into a structural prior that modulates physical prediction. On 2D airfoil and 3D car tasks, GeoABC consistently adapts to multiple neural operator backbones, reducing near-boundary relative $L_2$ error by $\sim$38% on average, narrowing the structural near-wall gap shared by mainstream neural operators, and advancing neural operators toward high-fidelity aerodynamic simulation.

Scientists have historically relied on mathematical models based on differential equations to relate system inputs -- forces, fluxes, or heat sources -- to outputs, such as displacement, velocity, concentration, and temperature. These models rely on deep domain knowledge to determine the form of the governing differential equation, which is then calibrated with data by solving an inverse problem. In recent years, the field of Scientific Machine Learning has introduced a variety of alternative modeling strategies for physical systems. A method called Sparse Identification of Nonlinear Dynamics learns the governing equation as a sparse linear combination of terms in a user-defined library. Neural Ordinary Differential Equations construct the governing equation by taking in the state and its derivatives at the input layer of a neural network. Entirely foregoing the modeling framework of differential equations, neural operators directly learn a non-linear mapping between the system inputs and outputs. From inverse problems to neural operators, all of these modeling strategies can be conceptualized as data-driven machinery to predict a system's response over a range of inputs. It is then natural to wonder how exactly these various strategies relate to each other, and whether they can be neatly taxonomized. Drawing from the philosophical literature on scientific models, we argue that many model types have a common structure, differing only in the assumed model class of the input-output relation they define. Connecting to philosophical ideas on mechanism, and arguing that data from physical systems arises from solutions to parsimonious differential equations, we propose that only certain models are capable of mechanism discovery, and thus generalization. Our analysis is intended to unite apparently disparate modeling strategies and provide insight into their appropriate use cases.

Neural operators such as the Fourier Neural Operator (FNO) have emerged as powerful surrogates for solving partial differential equations (PDEs), achieving speedups of several orders of magnitude over traditional numerical solvers. However, deploying these models in safety-critical engineering applications -- such as thermal management of electronic components and battery systems -- requires not only accurate point predictions but also rigorous uncertainty guarantees. Existing uncertainty quantification (UQ) methods for neural operators, including Monte Carlo Dropout and Deep Ensembles, provide only relative uncertainty estimates without formal coverage guarantees. In this work, we propose the first application of split conformal prediction to neural operator-based physics simulation, providing distribution-free prediction intervals with finite-sample coverage guarantees. We further introduce a normalized conformal prediction scheme that leverages MC Dropout uncertainty to produce adaptive-width intervals, yielding tighter intervals in regions of low uncertainty and wider intervals where the model is less certain. Full-scale experiments (33.7M parameters, 800 training samples, 5 ensemble members, NVIDIA V100) on steady-state heat conduction benchmarks demonstrate that our method achieves 89.1% empirical coverage at the target level of alpha=0.1, while producing spatially adaptive prediction intervals that reflect the underlying physical uncertainty structure. We also provide an uncertainty decomposition framework that separates epistemic uncertainty (68% of total) from aleatoric uncertainty (32% of total), offering actionable guidance for data collection and model improvement. Our method is implemented in an open-source platform with REST API endpoints and interactive 3D visualization.

We introduce GENERIC-FNO, the first neural operator to embed the full GENERIC (metriplectic) structure of nonequilibrium thermodynamics -- reversible, energy-conserving dynamics and irreversible, entropy-producing dynamics coupled through the degeneracy conditions -- directly in function space. Existing structure-preserving neural operators enforce at most a single conservation law or reversible (Hamiltonian) structure, while thermodynamically consistent learning has been confined to finite-dimensional, graph, or particle systems. GENERIC-FNO closes this gap: it learns the energy and entropy functionals as neural operators and parameterizes the Poisson and friction operators as diagonal Fourier multipliers sandwiched between rank-one projections that enforce the degeneracy conditions exactly, by construction, with no penalty term, update projection, or residual. The degeneracy identities hold to machine precision (residuals ~10^-13) for any initialization, dimension, or resolution, so the continuous-time dynamics conserve the learned energy and produce entropy exactly; the explicit time stepping adds only a small O(dt^2) drift (per-step residual ~10^-6). We further note that the (E,S,L,M) decomposition of a given flow is not unique, and introduce a gauge-invariant dissipation diagnostic separating reversible from dissipative dynamics independently of the learned functionals. Across three operator backbones (1D/2D FNOs and DeepONet) and four PDEs spanning reversible, dissipative, and mixed regimes, GENERIC-FNO preserves its exact structural guarantees zero-shot across a 4x super-resolution range (64 to 256), recovers the ground-truth ordering of physical dissipation, and is competitive with strong unconstrained and energy-penalized baselines, outperforming them on several dissipative and mixed problems at comparable or fewer parameters.

Physics-informed neural operators (PINOs) aim to learn solution operators for partial differential equations by using the governing physics as supervision, rather than relying solely on paired input-output simulation data. By incorporating physical constraints into the training objective, PINOs combine the cross-instance generalization of neural operators with the data efficiency of physics-informed learning. Despite this promise, how to train PINOs efficiently and robustly remains less well-understood than the training of either data-driven neural operators or physics-informed neural networks (PINNs). To bridge this gap, we examine key components of the PINO training pipeline, including architecture design, optimizer choice, loss balancing, and collocation-point sampling strategy. We study three representative operator backbones, Deep Operator Network (DeepONet), Fourier Neural Operator (FNO), and Continuous Vision Transformer (CViT), across five diverse parametric PDE systems. Our results show that CViT provides consistently strong and stable performance across the considered benchmarks. Beyond architecture, we find that several optimization pathologies previously identified in PINN training naturally arise in PINOs, including gradient conflicts and causal violation. We also find that mitigation algorithms developed for PINNs remain effective in the PINO setting. We further compare physics-informed and data-driven training under different data regimes, revealing that a carefully designed physics-informed training pipeline can match, and in some cases, outperform purely data-driven neural operators. Taken together, these findings provide a systematic empirical understanding of the optimization challenges in PINO training and inform a practical pipeline for efficient and robust physics-informed operator learning. Code and data are available at https://github.com/NanxiiChen/PI-CViT.

Neural operator surrogates (NO) approximate PDE solutions orders of magnitude faster than numerical solvers, but suffer from spectral bias: high-frequency content is systematically attenuated, limiting reliability where fine-scale structure matters. Sparse sensor measurements of the field are often available too, offering pointwise accuracy without spectral distortion but covering only a small fraction of the domain. We address this by treating NO predictions as auxiliary observations in a diffusion posterior sampling framework. Our method, FreqNO-DPS (https://github.com/niccoloperrone/FreqNO-DPS), combines an unconditional score-based diffusion prior, trained on high-fidelity simulations, with diffusion posterior sampling (DPS) conditioned on sparse observations and guided by a frozen neural operator. Naive integration reintroduces the surrogate's spectral bias; we resolve this with a closed-form, spectrally shaped guidance score that weights the surrogate by its frequency-dependent accuracy and needs no denoiser backpropagation. A distribution-free analysis bounds the approximation error across the frequency-diffusion-time plane and shows the guidance's frequency dependence is preserved regardless of distributional assumptions. On 3D elastic wavefield prediction at 5% and 2% sensor coverage, the method reaches near-zero spectral bias across all bands, where both the surrogate and sensor-only DPS show systematic high-frequency attenuation. Isotropic guidance, the natural baseline, improves pointwise accuracy but carries the bias into the posterior nearly intact, confirming that frequency-dependent calibration is essential, not merely beneficial. The framework needs only paired surrogate/reference data and exploits no problem-specific structure beyond the residual's approximate spectral diagonality, verifiable for new surrogates via the coherence diagnostic we provide.

Artificial-intelligence surrogates can support second-by-second thermal-hydraulic forecasting, but models selected and frozen offline may become condition-locked once deployed outside their pretraining envelope. This study develops a guarded continual-adaptation framework for experimental thermal-hydraulic loop data in which role-separated agents - Monitor, Diagnosis, Adaptation, Safety-Auditor, and Orchestrator - diagnose error signatures, prioritize candidate model families, and review promotions, while deterministic champion-challenger gates and background shadow learning retain final authority over model replacement. Seven surrogate families were screened by blocked three-fold cross-validation, and a temporal Fourier neural operator was selected as the initial champion for 60-s-history-to-10-s-trajectory forecasting on two held-out transients, with three seeds per adaptive mode. Static deployment gave a channel-averaged MAE of 7.06 and a 56.8% warning-exceedance ratio; rule-based adaptation reduced MAE to 6.54, whereas shadow refresh alone remained close to Static. The MA-Full mode, in which the role-separated multi-agent council reviews every evaluated stream step, achieved the lowest mean error, 5.72, and 35.8% exceedance, corresponding to a 19.0% improvement over Static. Paired bootstrap intervals against Static excluded zero, although intervals among adaptive modes overlapped and the six paired units limit broad statistical claims. Validated promotions from the neural operator to Transformer and graph neural network indicate that logged, gate-controlled adaptation can support auditable surrogate evolution while deterministic gates retain deployment authority.

Neural operators learn mappings between infinite-dimensional function spaces and provide a data-driven surrogate modeling paradigm for parametric partial differential equations (PDEs). Existing architectures typically obtain expressivity by parameterizing integral kernels in prescribed transform domains or by applying attention-like interactions over discretized spatial points. While these approaches have achieved substantial progress, they often face a persistent trade-off among physical interpretability, nonlocal spatial communication, mesh scalability, and computational cost. We propose a Light-inspired neural operator(LiNO), an operator-learning architecture whose latent evolution is decomposed into three mechanisms motivated by elementary light transport: reflection, refraction, and scattering. Reflection and refraction act as adaptive pointwise transformations in latent feature space, enabling local feature reorientation and anisotropic modulation, whereas scattering performs input-dependent nonlocal propagation over the physical domain. We first formulate scattering as a normalized pairwise kernel with relative positional bias, and then develop an efficient scattering variant that replaces explicit pairwise interactions with positive-feature global propagation and a local diffusion branch, reducing the dominant spatial complexity from quadratic to linear. This yields a structured neural operator that separates local feature modulation from global spatial communication while retaining a modular and interpretable latent evolution.

Deep learning surrogates for 3D Partial Differential Equations (PDEs) often fail to generalize across geometric transformations because they depend heavily on specific coordinate systems. While equivariant networks offer a solution, they typically rely on local operations in the spatial domain, making the global receptive field, which is essential for PDE dynamics, computationally expensive. Conversely, Fourier Neural Operators (FNOs) efficiently capture global interactions, yet establishing 3D equivariance within them remains impractical due to the prohibitive cost of spectral group convolutions. To bridge this gap, we introduce EqGINO, a geometrically robust framework that enforces isotropy in the spectral domain. By design, EqGINO guarantees exact equivariance to the discrete symmetries inherent to the discretized computational domain. Beyond this discrete guarantee, our structural prior enables effective generalization to arbitrary continuous orientations even with a limited number of SE(3)-transformed training samples. Consequently, our method robustly models coordinate-invariant physical laws on complex irregular 3D geometries. Our code is available at https://github.com/sung-won-kim/EqGINO

Existing evaluations of neural operators and in-context operator learning rely primarily on prediction error, but accurate output prediction does not guarantee the correct local dynamical structure. A model may match solutions while exhibiting incorrect sensitivities, distorted frequency response, spurious mode coupling, or unstable tangent behavior. We introduce a Jacobian-based spectral audit for in-context operator learning. For a fixed prompt, we differentiate the network output with respect to the query function and view the resulting Jacobian as a learned tangent operator. Projecting it onto Fourier modes, we obtain a local spectral characterization of the inferred operator, including frequency-dependent gains, phase structure, and cross-mode coupling. The audit complements standard prediction metrics by testing whether the model reproduces local mechanisms of the underlying PDE operator rather than only outputs. Across benchmarks, the audit reveals distinct operator-level phenomena, including phase transport, viscosity-dependent damping, nonlinear mode coupling, and reaction--diffusion stability structure. It also detects failures partially hidden by prediction-error metrics, including high-frequency degradation, incorrect phase recovery, and prompt--operator inconsistencies. Corrupted or internally inconsistent prompts lead to degraded tangent-operator structure even when pointwise predictions remain partially accurate. Our results suggest that prediction accuracy and local operator fidelity are distinct properties of learned neural operators. Our framework also provides a diagnostic for stability, sensitivity, and operator consistency.

Methane is a major driver of near-term climate change, and rapidly identifying its emission sources is a critical climate intervention. Spaceborne hyperspectral imagery is the primary tool for this task, but the volume of data produced by each sensor makes ground-based detection impractical and necessitates onboard detection. Classical methods incur prohibitive computational cost on onboard hardware, while deep learning models are fast but fall short on detection quality. We propose FLAME, a physics-guided neural operator that builds the physics of methane absorption directly into its architecture. On the methane detection benchmark, FLAME achieves the highest detection accuracy among all evaluated methods, reduces the pixel-level false positive rate by nearly $3\times$ over the strongest neural baseline, uses the fewest parameters among learned baselines, and runs within the latency budget of onboard satellite hardware.

Inspired by the function-space theory of neural networks, we formulate and analyze a variation space for nonlinear operators between Hilbert spaces, defined through vector-valued Borel measures of bounded variation. We characterize its unit ball as the closed convex hull of a vector-valued single-neuron dictionary in Bochner spaces. For the ReLU activation, the bounded linear operators in this space are precisely the Schatten-$1$ operators, with equivalent norms. For operators in this space, we establish encoder--decoder approximation bounds in the Bochner $L^q$-norm, where the error decomposes into input and output encoding errors and a finite-width term of order $N^{-1/2}$. Under sub-Gaussian assumptions on the input and noise, we further derive high-probability generalization bounds for empirical least squares over path-norm-constrained encoder--decoder networks; the finite-sample contribution to the squared prediction error is of order $K^{-1/2}$ up to logarithmic factors. The finite-width and finite-sample constants are independent of the encoding dimensions and bases, with the latter also independent of the network width. When the encoding errors decay algebraically, these bounds yield algebraic approximation and learning rates, in contrast to the complexity barriers for Lipschitz and Fréchet differentiable operator classes.

Neural operators provide fast surrogate models for PDE simulations, but standard architectures often treat geometry and discretization as secondary to field data. Physical states are usually represented as grid-channel stacks, even when different quantities naturally belong on vertices, edges, faces, cells, boundaries, or interfaces and must satisfy compatibility constraints. We propose Cellular Sheaf Neural Operators, a discretization-aware framework for structure-preserving neural PDE surrogates. The method represents PDE states on oriented cell complexes, couples local feature spaces through learned restriction maps, and uses incidence/Hodge-informed message passing to follow computational geometry. Learned update heads pass through coboundary or flux maps, allowing selected constraints to arise from cell-complex structure rather than only from loss penalties. For magnetohydrodynamics, this yields face-based magnetic-flux updates driven by edge electromotive fields and finite-volume-style fluid updates driven by learned face fluxes and cell sources. On turbulent MHD and fusion-equilibrium surrogate tasks, the method improves structure-sensitive diagnostics, including rollout behavior, divergence control, spectral error, and equilibrium-regression accuracy. These results indicate that cellular-sheaf structure is a useful inductive bias for neural PDE surrogates in constrained multiphysics systems.

We develop a comprehensive mathematical and computational framework for sequential surrogate modeling of three-phase black-oil reservoir dynamics using neural operators, with particular emphasis on Fourier Neural Operators (FNO) and their physics-informed variant (PINO). The application focus is the Norne benchmark reservoir, defined on a heterogeneous $46\times112\times22$ grid ($N=113,344$ cells), with a production history spanning $T=30$ timesteps covering 3298 days. Our theoretical contributions are organized around four interlocking problems: (1) functional-analytic formulation in a product-Sobolev-space setting, including well-posedness of the implicit timestep map and sharp local Lipschitz estimates; (2) covariate shift quantification, proving that the Wasserstein-2 distance grows as $W_2 \leq \varepsilon(L^n-1)/(L-1)$, with exponential population-risk discrepancy for $L>1$; (3) physics-constrained spectral stability, showing PINO training with $λ_R \geq λ^*_R$ reduces the learned Jacobian spectral radius to $ρ_F + Cλ_R^{-1/2}$, yielding uniform-in-time rollout error $|δ_n| \leq \varepsilon/(1-ρ)$; and (4) $K$-step TBPTT gradient analysis, deriving geometric bias decay $O(ρ^K)$, optimal window $K^ = O(\log(T/σ^2))$, and Adam convergence $O(1/\sqrt{t}) + O(ρ^{K^*})$. Empirical validation confirms all theoretical predictions: autoregressive PINO surrogates sustain $R^2>0.99$ (oil), $R^2>0.90$ (gas), $R^2\approx 0.80$ (pressure), and monotonically improving $R^2$ (water) across the full 3298-day horizon, trained on eight NVIDIA B200 GPUs in under one hour. A 1000-member ensemble runs in under one minute on a single B200 GPU, giving a ${\sim}10^4\times$ wall-clock speedup over the OPM finite-volume simulator.

Learning-based surrogates have become increasingly effective for wave-field prediction, and neural operators in particular have shown strong performance within observed frequency regimes. However, higher-frequency prediction under scarce target supervision remains comparatively underexplored, especially in wave problems where higher-frequency data are substantially more expensive to simulate or measure than lower-frequency data. A central difficulty is that cross-frequency transfer is inherently asymmetric: coarse amplitude structure remains relatively stable across frequencies, whereas phase-sensitive oscillatory structure deteriorates much more rapidly as frequency increases. Motivated by this asymmetry, we propose APEX, Amplitude-anchored and Phase-prior-guided Enhancement from eXtrapolated coarse predictions, a framework for target-scarce higher-frequency wave-field prediction. A lower-frequency neural operator first provides a coarse prediction in the target-frequency regime, from which we retain only the amplitude as a transferable structural anchor. A conditional flow-matching enhancer then reconstructs the target higher-frequency field under the guidance of a Green's-function-inspired phase prior. Experiments on SimpleWave, Helmholtz, and Maxwell benchmarks show that APEX consistently outperforms direct lower-to-higher extrapolation, target-adapted operator, and joint generative baselines under limited target-frequency supervision. Our results suggest that reliable higher-frequency prediction of oscillatory wave fields should not rely on direct end-to-end transfer of the full complex field, but instead on explicitly reusing transferable coarse structure while separately recovering the missing oscillatory detail.

Efficient onboard multi-field sparse reconstruction is essential for the autonomous operation of aerospace vehicles. While existing deep learning models exhibit promise for single-field reconstruction, deploying multiple independent models leads to prohibitive model size growth and fails to exploit cross-field correlations, particularly under few-shot conditions. To address these challenges, we first propose a lightweight multi-task Fourier neural operator (MTL-FNO), an end-to-end joint training framework based on hard parameter sharing. In each layer, the parameters are divided into shared and task-specific components to capture common features across fields while preserving task-specific characteristics. Moreover, the task-specific fine-tuning parameters are implemented as low-rank terms, achieving substantial model compression. Second, to address the difficulty of co-optimizing shared and task-specific parameters along with their real and imaginary parts, we revisit the FNO's spectral weight from a polar-form perspective and devise a physically meaningful decoupled optimization scheme. Specifically, we apply polar decomposition to slice-wise disentangle the spectral weight into a unitary tensor encoding phase information and a positive semi-definite tensor characterizing amplitude. By decoupling the optimization of phase and amplitude, our method can effectively mitigate tasks conflict. Meanwhile, to preserve unitary geometric fidelity during training, the Cayley transform is introduced to reparameterize the unitary tensor, converting the constrained optimization problem to an unconstrained one. Finally, the effectiveness of the proposed method under few-shot conditions is validated on two representative engineering cases. Results show that MTL-FNO achieves accuracy comparable to or even surpassing that of standard FNO, while reducing total model size by 76% and 60%, respectively.

Bayesian inverse design provides a principled framework for inferring aerodynamic geometries from sparse flow observations while quantifying uncertainty. However, its practical use in computational fluid dynamics (CFD) is severely limited by the cost of repeated high-fidelity simulations required for gradient-based Markov chain Monte Carlo (MCMC) sampling. While surrogate models are commonly proposed to reduce this cost, their effect on posterior geometry and uncertainty, especially for shock-dominated flows, remains poorly understood. In this work, we demonstrate that neural operator surrogates can be embedded directly within the MCMC inference loop while preserving posterior structure. Using a fully Bayesian inverse formulation of quasi-one-dimensional nozzle flow, we demonstrate that geometry parameterization plays a decisive role in identifiability and posterior conditioning, with cubic B-splines yielding stable and physically meaningful uncertainty estimates. Building on this formulation, a Deep Operator Network trained on CFD-generated data is substituted for the CFD solver within a No-U-Turn Sampler, while keeping the likelihood model, priors, and sampling configuration unchanged. Across sparse to fully observed regimes, surrogate-based inference reproduces the posterior geometry and uncertainty trends of the CFD reference. As a result of surrogate integration, total inference time is reduced to under one second, corresponding to a speedup exceeding three orders of magnitude. In addition, a direct inverse neural operator is examined as a deterministic alternative for inverse design, enabling single-shot geometry reconstruction without posterior sampling. These results demonstrate that neural operator-accelerated Bayesian inference enables practical, uncertainty-aware inverse design workflows for aerodynamic applications.

Neural operators learn mappings from function-dependent inputs to solutions, providing an effective framework for solving partial differential equations (PDEs). For time-dependent PDEs, existing methods typically perform long-horizon prediction through autoregressive rollout directly in high-dimensional physical field spaces, where each predicted state is recursively fed back as the input for the next step. Although effective for short-term prediction, this autoregressive rollout and the lack of continuous-time modeling lead to progressive error accumulation over long-horizon rollouts. In this work, we propose Autoregression-Free Neural Operators (AFNO), which map the time evolution of PDEs into a latent space and model continuous-time vector fields within it. AFNO uses flow matching to learn the latent vector field, thereby enabling continuous evolution over extended horizons, avoiding autoregressive rollout and capturing dynamics under varying parameter configurations through explicit conditioning on physical parameters. Theoretical analysis and extensive experiments on six PDEs demonstrate that AFNO improves long-horizon prediction stability and consistently reduces rollout errors compared with the baselines.

This work introduces the Wavelet-Laplace Neural Operator (WLNO), a novel neural operator that fuses Haar wavelet multi-scale spatial decomposition with the Laplace-domain pole-residue formulation of the Laplace Neural Operator (LNO). While LNO captures transient and steady-state dynamics through learnable system poles and residues, it lacks an explicit mechanism for extracting spatially localized multi-scale features inherent in complex PDE solutions. WLNO addresses this by augmenting the LNO core with a parallel single-level Haar discrete wavelet transform (DWT) branch that decomposes the lifted feature map into four frequency subbands: approximation (LL), horizontal detail (LH), vertical detail (HL), and diagonal detail (HH) and applies independent learned $1\times1$ convolutions to each subband before reconstruction via the inverse DWT. The two branches are fused through a learnable sigmoid-gated weight $α_\mathrm{wav}$, initialized to give a small initial contribution to the wavelet branch, allowing the model to adaptively balance Laplace-domain dynamics against spatial multi-scale features throughout training. WLNO is evaluated against LNO on five benchmark PDE problems using identical hyperparameters, training data, and evaluation protocols: the diffusion equation, the Burgers equation, the reaction-diffusion system, Darcy flow, and the two-dimensional Navier-Stokes equation. WLNO consistently outperforms LNO on all five problems, with the most pronounced improvement on problems with strong spatial multi-scale structure, such as the Burgers equation with sharp shock fronts and the Navier-Stokes equation with coherent vortical structures, while remaining consistent across smoother and elliptic problems. These results demonstrate that wavelet-based multi-scale spatial decomposition is a principled and effective complement to Laplace-domain operator learning.

We propose a Weak-form Physics-Informed Neural Operator (WINO), a data-free framework that combines the efficiency of neural operators with the geometric flexibility of the $\varphi$-finite element method ($\varphi$-FEM). $\varphi$-FEM is an unfitted method that accommodates geometric variations without body-fitted meshes, where the domain geometry is represented by the level-set function $\varphi$. To impose the boundary conditions, Dirichlet problems adopt the $\varphi$-FEM lifting so only the homogeneous displacement contribution is learned, whereas traction-driven Neumann problems additionally predict the auxiliary fields necessary for the unfitted weak formulation. Parameters are trained by minimizing squared weak-form residuals aligned with $\varphi$-FEM together with squared penalties on the cut-cell auxiliary equations, which removes the need for large paired datasets of converged reference solutions. After training, WINO outputs can seed the nonlinear $\varphi$-FEM solvers as neural operator warm starts (NOWS), which reduce iteration counts relative to traditional cold-started solvers. Numerical benchmarks show that WINO achieves high accuracy below 0.04 across all benchmarks, while reducing total computational time by 50--80% compared with purely data-driven methods.

The eigendecomposition of the Laplace--Beltrami Operator (LBO) is fundamental to geometric analysis, yet computing its low-frequency eigenmodes remains a significant bottleneck due to the high cost of iterative solvers on large-scale data. To amortize this cost, we introduce the Neural Eigenspace Operator (NEO), a feed-forward framework designed to predict the spectrum directly from point clouds. Crucially, NEO circumvents the ill-posed nature of standard eigenvector regression, which suffers from intrinsic sign flips and rotation ambiguities, by learning the stable, invariant low-frequency subspace instead. Specifically, the network predicts a redundant set of basis functions whose span robustly covers the target eigenspace, allowing for the recovery of accurate eigenpairs via a lightweight Rayleigh--Ritz refinement. To handle irregular sampling, we propose a mass-aware neural operator that incorporates per-point area weights into attention-based aggregation, improving robustness to non-uniform densities and enabling zero-shot generalization across resolutions. Our approach achieves near-linear runtime scaling and substantial wall-clock speedups over iterative solvers at comparable accuracy, and exhibits strong zero-shot transfer to high-resolution point clouds. The resulting eigenpairs support standard spectral geometry tasks, while the raw basis functions provide effective point-wise features for downstream learning. Code: https://github.com/Adversarr/NEO.

Defocus deblurring in pathological microscopy remains challenging due to the spatially varying and locally discontinuous nature of optical blur induced by a position-dependent integral imaging process. Existing deep learning methods, constrained by shift-invariance assumptions and limited interpretability, are not well suited to such heterogeneous blur patterns. Neural operators provide a principled alternative by modeling defocus formation directly as an integral operator, offering a new perspective on defocus deblurring. However, most existing neural operator architectures for low-level vision rely on globally parameterized kernels that assume smoothness and stationarity, limiting their ability to model heterogeneous and locally discontinuous blur patterns. To address this limitation, we propose the Discontinuous Galerkin Neural Operator (DGNO), which parameterizes the integral kernel using a discontinuous Galerkin formulation with element-local volume operators and interface numerical fluxes. DGNO provides a principled combination of locality, heterogeneity modeling, and global coherence while preserving the underlying physics of optical image formation. Extensive and insightful experiments demonstrate that DGNO surpasses state-of-the-arts, delivering sharper reconstructions, robust handling of spatially varying blur, and scalable high-resolution performance. The code will be released at https://github.com/DeepMed-Lab-ECNU/Single-Image-Deblur.

We study the approximation and statistical complexity of learning collections of operators in a shared multi-task setting, with a focus on the Multiple Neural Operators (MNO) architecture. For broad classes of Lipschitz multiple operator maps, we derive near-optimal upper bounds for approximation and statistical generalization. On the lower-bound side, we establish a curse of parametric complexity and prove corresponding minimax rates. Together, these results show that shared representations across tasks do not increase the overall cost: multi-task operator learning follows the same scaling laws as single operator learning. We also compare MNO with a multi-task extension of DeepONet based on concatenated task inputs and show that, from a worst-case approximation-complexity perspective, both architectures satisfy essentially the same asymptotic rates.

We introduce an abstract neural flow framework for neural networks and neural operators. The framework contains two continuous-depth models, namely neural flows with composition and separation structures, and covers both finite-dimensional function approximation and infinite-dimensional operator approximation. We prove well-posedness and universal approximation properties for the corresponding neural flows, including, to the best of our knowledge, the first universal approximation result for flow-based models between infinite-dimensional spaces. We also obtain universal approximation results for convolutional neural flow models. Through suitable time discretizations, the composition structure recovers ResNet-type architectures, while the separation structure, via a splitting-based discretization, yields plain architectures. This gives a unified flow-based route to both residual and plain architectures for neural networks and neural operators with fully connected or convolutional linear layers.

Neural operators have achieved significant success in modern scientific computing due to their flexibility and strong generalization capabilities. Existing models, however, primarily rely on first-order kernel integral approximations, which severely limit their expressivity. To address this, we propose the Infinite-order Kernel Neural Operator (IKNO), which constructs neural operators via infinite-order kernel integrals and admits an elegant closed-form finite approximation. We develop two complementary infinite-order neural operator constructions: IKNO-Vanilla, which applies the full-kernel resolvent on the product grid via Kronecker eigendecomposition, and IKNO-TP, an alternative tensor-product operator that composes per-axis resolvents. Furthermore, we develop fast computation schemes for both variants of IKNO, which achieve outstanding global information aggregation while maintaining high computational efficiency. Empirically, we evaluate our IKNO on both time-dependent and time-independent benchmarks with arbitrary input shapes, including large-scale industrial datasets. Extensive experiments demonstrate that the IKNO method consistently achieves the SOTA accuracy with significant improvements on nearly all benchmark datasets while maintaining scalability to very large point clouds.

Solving partial differential equations with neural operators significantly reduces computational costs but remains bottlenecked by high training data requirements. Active learning offers a natural framework to mitigate this by selectively acquiring the most informative samples in an iterative manner. We introduce physics-based acquisition - a novel physics-informed active learning algorithm that leverages the partial differential equation residual to guide data selection. We validate the method by presenting numerical experiments for the 1D Burgers equation and the 2D compressible Navier-Stokes equations. We show that, in our experiments, physics-based acquisition consistently outperforms random acquisition and matches the state of the art in data efficiency. At the same time, it has the unique advantage of injecting a physics inductive bias into the training process, ensuring that simulation cost is spent where the model's physical understanding is weakest.

Neural operators have achieved strong performance in learning solution operators of partial differential equations (PDEs), but their inherently continuous representations struggle to capture discontinuities and sharp transitions. Existing approaches typically approximate such features within continuous function spaces, often requiring increased model capacity and high-resolution data. In this work, we propose Cut-DeepONet, a two-stage training framework that explicitly models discontinuities while reducing learning complexity. Our approach reformulates the problem via a lifting strategy, partitioning the domain into smooth subregions while representing discontinuities as boundaries in a higher-dimensional space. This separation aligns the operator learning task with the inductive bias of neural networks and avoids directly approximating discontinuities. An additional network predicts input-dependent discontinuity locations for unseen inputs, which are then used to guide the neural operator in generating smooth components within each region. Experiments on benchmark PDEs show that Cut-DeepONet outperforms state-of-the-art methods, even when trained on low-resolution datasets. The method excels on problems with discontinuities and sharp transitions, while using fewer trainable parameters. Our results highlight the benefits of changing the representation of operator learning rather than increasing model complexity.

ANN-to-SNN conversion offers a practical, training-free route to spiking large language models. However, current pipelines primarily focus on spike-driven realizations for Transformer linear-algebra operations, while providing limited support for key nonlinear operators. This gap limits compatibility with neuromorphic-style execution constraints, where such nonlinearities typically require division, exponentiation, or norm computations that are not naturally supported by standard leaky integrate-and-fire dynamics. To solve this problem, we propose a plug-and-play framework that implements spike-friendly approximations for Transformer nonlinearities and integrates into existing ANN-to-SNN pipelines. Our method decomposes these nonlinear computations into three recurring primitives -- division, exponentiation, and $\ell_2$ norms -- and realizes them via population computation using LIF neuron groups, combined with lightweight bit-shift scaling to avoid floating-point arithmetic. By composing these primitives as modular operator blocks, our framework supports common Transformer nonlinearities (e.g., Softmax, SiLU, and normalization) without any fine-tuning. Experiments on a range of LLMs Transformers show that selectively replacing the targeted nonlinear operators incurs less than a $1\%$ accuracy drop across all evaluated tasks.

Continuum robots enable dexterous manipulation in constrained environments, but require accurate and efficient models for real-time manipulation and control. Traditional physics-based models can be computationally expensive and may suffer from inaccuracies due to unmodeled effects, while current learning-based methods often generalize poorly beyond the specific robot on which they are trained. We present a formulation of surrogate modeling for tendon-driven continuum robots as an operator learning problem that maps robot design parameters and tendon actuation inputs to resulting configurations. This formulation enables a single trained model to generalize across a large class of robot designs. We develop four novel neural operator architectures--two based on Deep Operator Networks (DeepONets) and two based on Fourier Neural Operators (FNOs)--and train them on simulation data to predict robot configurations. All architectures achieve good accuracy while allowing for fast and accurate generalization across designs. Our results demonstrate that operator learning provides an effective and generalizable surrogate for continuum robot mechanics in the design space, enabling fast modeling for control, planning, and design optimization in surgical and industrial applications.

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.

The modeling and simulation of infinite-dimensional Hamiltonian systems are central problems in mathematical physics and engineering, however they pose significant computational and structural challenges for standard data-driven architectures. In this work, we introduce the Symplectic Neural Operator, a neural operator architecture designed to preserve the symplectic structure intrinsic to Hamiltonian PDEs. We provide a theoretical characterization of their symplecticity and establish a rigorous long-term stability result based on the combination of symplectic structure preservation and learning accuracy. Numerical experiments on canonical Hamiltonian PDEs corroborate this theoretical result and show that SNOs exhibit improved energy behavior compared with non-structure-preserving neural operators.

Neural operators excel as deterministic surrogates, but inevitably collapse to the conditional mean when applied to stochastic PDEs, discarding the variance and tail structure upon which uncertainty quantification depends. Recovering this structure typically requires Monte Carlo rollouts or grafted generative models, both of which surrender the one-shot efficiency and resolution invariance that define the operator paradigm. To resolve this, we draw on the Doob-Meyer theorem, which establishes that any semimartingale fundamentally decomposes into a predictable drift and an unpredictable, zero-mean martingale. Translating this theorem into an architectural prior, we introduce the Martingale Neural Operator (MNO). MNO maps an initial condition directly to the conditional mean and covariance of the terminal law, parameterized by a drift-like mean and a low-rank factor $B_φ$ with $B_φ^\top B_φ$ positive semi-definite by construction. For our experiments, we use a Gaussian residual instantiation. Across 1D SPDEs, rough volatility, and 2D operator tasks, MNO reduces Wasserstein distance by up to $120\times$ on $φ^4$ field theory and $68\times$ on stochastic Burgers, evaluating $\sim 3\times$ faster than a conditional diffusion baseline at matched wall-clock training budgets. On 2D tasks, MNO is comparable to FNO on zero-shot resolution transfer and turbulent flow, while quasi-deterministic systems such as Gray-Scott remain a failure mode.

Pre-training neural operators on diverse partial differential equation (PDE) datasets has emerged as a promising direction for building general-purpose surrogate models in scientific machine learning. However, the inherent complexity and structural diversity of PDE solution operators make multi-PDE pre-training fundamentally challenging. Existing methods mainly address this by increasing model capacity, while leaving the target solution operators unchanged. Inspired by classical numerical analysis, we instead propose to transform complex and diverse solution operators into simpler, better-aligned forms that are easier to model jointly. Since the optimal transformation varies across PDE types, it must be adaptive and input-dependent, allowing a single neural operator to approximate an entire family of operators. We instantiate this idea as AOT-POT (adaptive operator-transformation for pre-training operator transformer), which expands hidden representations into multiple parallel streams, adaptively aggregates and redistributes them before and after each sub-layer, and mixes streams through Sinkhorn-projected doubly stochastic matrices for stable training. These mechanisms together reshape diverse solution operators into a unified form that can be effectively modeled by a single architecture. Empirically, AOT-POT achieves state-of-the-art performance on 12 PDE benchmarks with only 3% additional parameters, reducing relative L2 error by up to 77.6% (40.9% on average). Fine-tuning AOT-POT further reduces L2 error by up to 92% on in-domain PDEs and 89% on out-of-domain PDEs (unseen types during pre-training), demonstrating that adaptive operator transformation is an effective and complementary direction for advancing PDE foundation models beyond simply scaling model capacity.

In this paper, we study solution operators of physical field equations on geometric meshes from a function-space perspective. We reveal that Hodge orthogonality fundamentally resolves spectral interference by isolating unlearnable topological degrees of freedom from learnable geometric dynamics, enabling an additive approximation confined to structure-preserving subspaces. Building on Hodge theory and operator splitting, we derive a principled operator-level decomposition. The result is a Hybrid Eulerian-Lagrangian architecture with an algebraic-level inductive bias we call Hodge Spectral Duality (HSD). In our framework, we use discrete differential forms to capture topology-dominated components and an orthogonal auxiliary ambient space to represent complex local dynamics. Our method achieves superior accuracy and efficiency on geometric graphs with enhanced fidelity to physical invariants. Our code is available at https://github.com/ContinuumCoder/Hodge-Spectral-Duality

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.

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.

Neural operators have become an effective framework for learning mappings between function spaces, yet most existing architectures realize operators within a single representational domain, such as physical, spectral, or latent space. In this work, we introduce UFO (Domain-Unification-Free Operator), a cross-domain neural operator framework that realizes operators through adaptive, jointly conditioned interactions among representations defined on distinct domains. UFO enables discretization decoupling: the input function can be observed at resolutions or locations different from those used during training, while the solution can be queried at arbitrary output resolutions. Across four complementary benchmarks covering discontinuous inputs, irregular sampling with spectral mismatch, nonlinear dynamics, and stochastic high-frequency fields, UFO delivers accurate, robust, and physically coherent predictions under distribution shifts. These results establish cross-domain, phase-modulated realization as a powerful framework for discretization-decoupled neural operator learning.

Neural operators provide a framework for learning solution operators of partial differential equations (PDEs), enabling efficient surrogate modeling for complex systems. While universal approximation results are now well understood, approximation analysis specific to nonlinear reaction-diffusion systems remains limited. In this paper, we study neural operators applied to the solution mapping from initial conditions to time-dependent solutions of a generalized Gierer-Meinhardt reaction-diffusion system, a prototypical model of nonlinear pattern formation. Our main results establish explicit approximation error bounds in terms of network depth, width, and spectral rank by exploiting the Laplacian spectral representation of the Green's function underlying the PDE. We show that the required parameter complexity grows at most polynomially with respect to the target accuracy, demonstrating that Laplacian eigenfunction-based neural operator architectures alleviate the curse of parametric complexity encountered in generic operator learning. Numerical experiments on the Gierer-Meinhardt system support the theoretical findings.

Most dimensionality reduction methods treat data as discrete point clouds, ignoring the continuous domain structure inherent to many real-world processes. To bridge this gap, we introduce Neural Operator Function Embedding (NOFE), a domain-aware framework for continuous dimensionality reduction. NOFE learns function-to-function mappings via a Graph Kernel Operator, enabling mesh-free evaluation at arbitrary query locations independent of input discretization. We establish NOFE as approximation of sheaf-to-sheaf mappings, generalizing Sheaf Neural Networks to continuous domains. We evaluate NOFE across different datasets, comparing it against PCA, t-SNE, and UMAP. Our results demonstrate that NOFE significantly outperforms baselines in local structure preservation, achieving a local Stress of 0.111 compared to 0.398 for PCA, 0.773 for t-SNE, and 0.791 for UMAP for the ERA5 climate reanalysis dataset. NOFE also exhibits robust sampling independence, reducing the Patch Stitching Error by up to $20.0\times$ relative to UMAP (59.0 vs. 267.6 under regional normalization) and ensuring consistency across disjoint domain patches. While maintaining competitive global structure preservation (Stress-1: 0.379 vs. PCA's 0.268), NOFE resolves fine-grained structures and produces smooth, consistent embeddings that generalize across varying sample densities, addressing key limitations of discrete reduction methods.

Partial differential equations (PDEs) govern diverse physical phenomena, yet high-fidelity numerical solutions are computationally expensive and Machine Learning approaches lack generalization. While Scientific Foundation Models (SFMs) aim to provide universal surrogates, typical encoding-decoding approaches suffer from high pretraining costs and limited interpretability. In this paper, we propose Compositional Neural Operators (CompNO) for 2D systems, a framework that decomposes complex PDEs into a library of Foundation Blocks. Each block is a specialized Neural Operator pretrained on elementary physics. This modular library contains convection, diffusion, and nonlinear convection blocks as well as a Poisson Solver, enabling the framework to address the pressure-velocity coupling. These experts are assembled via an Adaptation Block featuring an Aggregator. This aggregator learns nonlinear interactions by minimizing data loss and physics-based residuals driven from governing equations. The proposed approach has been evaluated on the Convection-Diffusion equation, the Burgers' equation, and the Incompressible Navier-Stokes equation. Our results demonstrate that learning from elementary operators significantly improves adaptability, enhances model interpretability and facilitates the reuse of pretrained blocks when adapting to new physical systems.

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.

jNO (jax Neural Operators) is a JAX-native library for neural operators and foundation models with unified support for both data-driven and physics-informed training. Its core design is a tracing system in which domains, model calls, residuals, supervised losses, and diagnostics are written in one symbolic language and compiled into one optimization pipeline. This allows users to move between operator regression, mesh-aware residual evaluation, and PDE-constrained training without restructuring the surrounding code. jNO also supports multi-model compositions, fine-grained control at parameter level (model, optimizer, and learning rate), hyperparameter tuning, and JAX-native workflows for translated PDE foundation-model families. The source repository is available at https://github.com/FhG-IISB/jNO.

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.

Neural operators have emerged as powerful data-driven solvers for PDEs, offering substantial acceleration over classical numerical methods. However, existing transformer-based operators still face critical challenges when modeling PDEs on complex geometries: directly processing over massive mesh points is computationally expensive, while operating in raw discretization coordinates may obscure the intrinsic geometry where physical interactions are more naturally expressed. To address these limitations, we introduce the Charted Axial Transformer Operator (CATO), a geometry-adaptive and derivative-aware neural operator for PDEs on general geometries. Instead of applying attention directly in the physical coordinate system, CATO learns a continuous latent chart that maps mesh coordinates into a learned chart space, where chart-conditioned axial attention efficiently captures long-range dependencies with reduced computational cost. In addition, CATO introduces a derivative-aware physics loss for steady-state PDEs that jointly supervises solution values, mesh-consistent gradients, and an auxiliary flux-like field, improving physical fidelity and reducing oversmoothing. We further provide a theoretical approximation result showing that, under a favorable chart, charted axial attention can represent low-rank axial solution operators with controlled error, and that small chart perturbations induce bounded approximation degradation. CATO achieves the best performance across all evaluated datasets, yielding an average improvement of approximately 26.76% over the strongest competing baselines while reducing the number of parameters by 81.98%. These results highlight the effectiveness of learning geometry-adaptive charts and derivative-aware physical supervision for accurate and efficient PDE operator learning.

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.

Physics-informed operator learning is an attractive candidate for surrogate modeling of microstructures, especially in multiscale finite-element simulations. Its practical use, however, is often limited by the high cost of loss evaluation. We address this bottleneck by combining the Equilibrium Neural Operator (EquiNO) with the QR-based discrete empirical interpolation method (Q-DEIM). EquiNO learns only the modal coefficients of reduced displacement-fluctuation and first Piola-Kirchhoff stress representations built from periodic and divergence-free bases, thereby enforcing periodicity and mechanical equilibrium by construction. Q-DEIM then identifies a small set of spatial points through a column-pivoted QR factorization of the stress basis and restricts constitutive evaluations during training to these points alone. This makes full-batch second-order optimization practical for three-dimensional representative volume elements (RVEs). Homogenized first Piola-Kirchhoff stresses are recovered directly from the offline-averaged reduced stress modes, without the need to reconstruct the full stress field at inference time. We validate the framework on two three-dimensional finite-strain hyperelastic RVEs. Q-DEIM reduces the per-step training cost by roughly three orders of magnitude relative to full-field loss evaluation, while reduced homogenization achieves speed-up factors of order $10^3$ to $10^4$ over direct full-field computations. Despite relying on only a small number of offline snapshot loading paths for basis construction, the method accurately interpolates and extrapolates both microscopic stress fields and homogenized stresses, with prediction quality improving systematically as more snapshots are added.

Normalization layers in neural operators usually compute statistics by uniformly averaging discrete grid values, making the normalization itself discretization-dependent and thereby a source of transfer error across different resolutions or meshes. To enable discretization robustness, we introduce a quadrature normalization family that replaces existing uniform averaging in normalization layers with numerical quadrature: QuadNorm and BlendQuadNorm. On endpoint-inclusive uniform grids, the proposed quadrature moments are $O(h^2)$-consistent across discretizations, meaning that their cross-resolution mismatch decays quadratically with grid spacing. A transfer-error bound then predicts how normalization-induced mismatch scales with both the resolution gap and network depth. The experiments show the same gap- and depth-scaling trends predicted by the transfer-error bound. On Darcy, QuadNorm delivers the best cross-resolution performance at every tested target resolution from $64^2$ to $256^2$; on real-data benchmarks, Transolver with QuadNorm achieves nearly resolution-invariant transfer. The largest gains appear on nonperiodic PDEs and nonspectral architectures, where native-resolution improvements also emerge. We also validate BlendQuadNorm, which stays close to LayerNorm behavior and serves as a conservative default for periodic FNO settings. These results identify normalization as a previously overlooked source of resolution dependence in neural operators.

Probabilistic conditioning is concerned with the identification of a distribution of a random variable $X$ given a random variable $Y$. It is a cornerstone of scientific and engineering applications where modeling uncertainty is key. This problem has traditionally been addressed in machine learning by directly learning the conditional distribution of a fixed joint distribution. This paper introduces a novel perspective: we propose to solve the conditioning problem by identifying a single operator that maps any joint density to its conditional, thus amortizing over joint-conditional pairs. We establish that the conditioning operator can be approximated to arbitrary accuracy by neural operators. Our proof relies on new results establishing continuity of the conditioning operator over suitable classes of densities. Finally, we learn the conditioning map for a class of Gaussian mixtures using neural operators, illustrating the promise of our framework. This work provides the theoretical underpinnings for general-purpose, amortized methods for probabilistic conditioning, such as foundation models for Bayesian inference.

Neural operators perform well on structured domains, yet their behaviour on irregular geometries remains poorly understood. We show that this limitation is not merely an encoding issue, but a depth-wise failure mode inherent to deep operator architectures. We formalise the Geometric Forgetting Hypothesis: due to the Markovian structure of operator layers and their reliance on global mixing mechanisms, neural operators progressively lose access to domain geometry as depth increases. Using layer-wise geometric probing, we demonstrate that both spectral and attention-based operators systematically lose geometric fidelity. We show that this geometric forgetting degrades accuracy, stability, and generalisation. To counteract it, we introduce a lightweight geometry memory injection mechanism that restores geometric constraints at intermediate depths with minimal architectural overhead. This simple intervention consistently mitigates forgetting and exposes a geometric shortcut instability in transformer-based operators, revealing that geometric retention is a structural requirement rather than a design choice.

We propose an architecture that augments the Flux Neural Operator (Flux NO), which combines the classical finite volume method (FVM) with neural operators, with ViT-based context injection. Our model is formulated as a hypernetwork: it extracts solution dynamics over a finite temporal window, encodes them with a recurrent Vision Transformer, and generates the parameters of a context-conditioned neural operator. This enables the model to infer and solve conservation laws without explicit access to the governing equation or PDE coefficients. Experimentally, we show that the proposed method preserves the robustness, generalization ability, and long-time prediction advantages of Flux NO over standard neural operators, while delivering reliable numerical solutions across a broad range of conservative systems, including previously unseen fluxes. Our code is available at https://github.com/xx257xx/CONTEXT_FLUX_NO.

Neural operators generally demonstrate strong predictive performance on in-distribution (ID) problems. However, a critical limitation of existing methods is their significant performance degradation when encountering out-of-distribution (OOD) data. To address this issue, this work introduces continual learning into physics-informed neural operators, with particular emphasis on neural operators built upon the Transolver architecture, and proposes a simple yet effective replay-based continual learning strategy. The proposed method is fully physics-informed and does not require labeled data, relying solely on input fields together with physical constraints for training. When new OOD data become available, a small number of past data are incorporated through a distillation-based constraint to preserve previously acquired knowledge and alleviate catastrophic forgetting. Meanwhile, a transfer learning LoRA is employed to enable rapid adaptation to the new data. The proposed framework is systematically validated on three representative physical problems, including the Darcy flow problem in fluid mechanics, a two-dimensional hyperelastic brain tumor problem in biomechanics, and a three-dimensional linear elastic Triply Periodic Minimal Surfaces problem in solid mechanics. The results demonstrate that the proposed method effectively mitigates catastrophic forgetting on previously learned data while maintaining fast adaptability to new data. Compared with conventional joint training strategies, the proposed method significantly improves training efficiency while reducing additional memory usage and computational cost.

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.

We introduce the Z-Domain Neural Operator (ZNO), a causal neural operator whose layers are stable low-rank multiple-input multiple-output (MIMO) rational filters parameterized directly in the $z$-plane. ZNO addresses a limitation of existing operator learning methods, many of which are primarily tailored for continuous-time problems, while a large class of system-identification problems is intrinsically discrete-time. The $z$-domain form expresses stability as a unit-disk pole constraint and makes learned discrete-time poles directly readable. The model combines low-rank channel mixing, smooth stable pole reparameterization, causal recurrence, and an optional short finite impulse response (FIR) branch in a single $z$-domain rational recurrent layer. Across controlled discrete system-identification experiments, ZNO's advantage is most evident when the target dynamics are stable rational systems with lightly damped poles near the unit circle. Under matched parameter budgets, ZNO is not uniformly dominant; however, with validation-selected configurations, the same architecture can achieve the lowest mean error across the controlled tasks. A five-bin difficulty sweep over near-unit-circle / long-memory dynamics shows that ZNO has the lowest mean error across memory regimes, from short (approximately 10 steps) to long (approximately 100-200 steps). On five public nonlinear system-identification benchmarks, ZNO is competitive with neural operator and state-space baselines, achieving the lowest mean error on benchmarks whose dynamics align with stable rational discrete-time filters, while classical or state-space baselines remain preferable on some systems. These results position ZNO as a strong model for stable rational discrete-time dynamics, especially in near-unit-circle and long-memory regimes, but not as a universal replacement for specialized system-identification methods.

Safety critical control of robotic manipulation tasks involving deformable media such as fluids, cloth, and soft objects remains challenging because existing learning based approaches encode safety indirectly through reward shaping, which provides no guarantee of constraint satisfaction at deployment. We present a constraint driven online safety filter for deformable object manipulation that enforces explicit task level safety constraints in real time by minimally modifying any nominal control policy. Our approach combines two key components: a horizon agnostic neural operator that learns the boundary input output mapping of the underlying PDE dynamics and generalizes across variable rollout lengths without retraining, and a boundary control barrier function that certifies safety at the task relevant output level via a lightweight quadratic program. The resulting safety constraint is affine in the boundary input rate, enabling real time online filtering. We evaluate the proposed method on fluid manipulation tasks in FluidLab, where the filter improves safe trajectory rates by up to 22% over unfiltered base policies while also reducing the number of steps required to reach the safe set, demonstrating that constraint driven safety enforcement is both more reliable and more efficient than reward shaping approaches.

We introduce HyCOP, a modular framework that learns parametric PDE solution operators by composing simple modules (advection, diffusion, learned closures, boundary handling) in a query-conditioned way. Rather than learning a monolithic map, HyCOP learns a policy over short programs - which module to apply and for how long - conditioned on regime features and state statistics. Modules may be numerical sub-solvers or learned components, enabling hybrid surrogates evaluated at arbitrary query times without autoregressive rollout. Across diverse PDE benchmarks, HyCOP produces interpretable programs, delivers order-of-magnitude OOD improvements over monolithic neural operators, and supports modular transfer through dictionary updates (e.g., boundary swaps, residual enrichment). Our theory characterizes expressivity and gives an error decomposition that separates composition error from module error and doubles as a process-level diagnostic.

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.