Multifidelity Operator Learning

• Multifidelity Operator Learning is the approach of approximating operators between function spaces using a blend of coarse and fine data from diverse simulation fidelities.
• It employs methodologies such as residual correction, reduced-order closures, and discretization-independent architectures to improve accuracy and reduce training costs.
• Applications include PDE closure, multiscale downscaling, and uncertainty quantification, with notable success in experiments on Burgers and Navier–Stokes equations.

Searching arXiv for recent and foundational papers on multifidelity operator learning to ground the article in the literature. Multifidelity operator learning studies how to approximate operators between function spaces when data or simulators are available at multiple levels of fidelity, cost, and resolution. In the recent literature, the target operator is typically a PDE solution map, a closure map for a reduced model, or a coarse-to-fine reconstruction map, while the low-fidelity information may come from coarse discretizations, reduced-order models, homogenized equations, diffusion limits, or partially converged solvers. The central objective is to exploit abundant inexpensive information together with scarce accurate information so that the learned operator approaches high-fidelity behavior without requiring a fully high-fidelity training set (Howard et al., 2022, Hauck et al., 9 Jul 2025).

1. Problem class and scope

In the standard formulation, an operator-learning problem seeks a map

$\mathcal{G} : \mathcal{X} \to \mathcal{Y},$

where the input is a function, field, or parametric specification and the output is another function or field. Multifidelity variants retain the same target, but assume that training data are distributed across multiple approximations of the same physical system. Typical pairings include coarse and fine discretizations, low- and high-order PDE models, reduced- and full-order dynamical models, and homogenized-to-fine or diffusion-to-transport maps (Thakur et al., 2022, Zhang et al., 2023).

This scope is broader than a simple “coarse grid plus fine grid” hierarchy. In reduced-order closure, the low-fidelity component can be a Galerkin POD ROM and the high-fidelity component can be the projection of a full-order model onto the same reduced space, so that the learned operator represents a missing-scale correction rather than a full surrogate (Ahmed et al., 2023). In homogenization-based downscaling, the low-fidelity input is a coarse or homogenized solution $u_0$, obtained analytically or via MsFEM, GMsFEM, NH-PINN, or a diffusion approximation, while the high-fidelity target is the fine-scale solution $u$ available only at sparse locations (Zhang et al., 2023). In parametric dynamical systems, the low-fidelity data can be coarsened and biased trajectories while the high-fidelity data are fine-grid solutions, and the learning task is to recover the high-fidelity operator with very few high-fidelity samples (Zheng et al., 1 Feb 2025).

A further conceptual refinement is the distinction between theoretical operator learning and its numerical realization. “Numerical operator learning” treats the trained model as a family of discretization-dependent realizations $\widetilde G_\theta^{(\mathcal S_U,\mathcal S_V)}$ of an underlying continuous operator $G_\theta$, while “discretization independence” measures whether the same parameters $\theta$ remain valid across many input and output discretizations (Hauck et al., 9 Jul 2025). This perspective makes multifidelity training especially natural, because changing fidelity is then understood as changing the discretization operators rather than changing the underlying operator.

2. Core mathematical formulations

The most common formulation is additive correction. If $u_L$ denotes a low-fidelity prediction and $u_H$ a high-fidelity output, then the high-fidelity operator is represented as a low-fidelity baseline plus a learned discrepancy,

$u_H = \mathcal{G}_L(\cdot) + \mathcal{R}(\cdot),$

or, in the nonlinear autoregressive form used in multifidelity neural operators,

$y_H = \mathcal{G}(y_L) + \delta(x).$

This residual viewpoint appears in multifidelity DeepONet, MF-WNO, and MF-LNO, and is the dominant mechanism for exploiting inter-fidelity correlation (Lu et al., 2022, Thakur et al., 2022, Zheng et al., 1 Feb 2025).

A second formulation is operator-valued correction in reduced coordinates. For projection-based ROM closure, the reduced state update is written as

$u_0$0

so that the learned operator maps low-fidelity reduced predictions to a closure correction. In the multifidelity DeepONet closure construction, the learned operator receives $u_0$1 and returns the residual $u_0$2 mode by mode (Ahmed et al., 2023).

A third formulation is coarse-to-fine downscaling. In homogenization-based operator learning, the target is

$u_0$3

or, at first-order asymptotic level,

$u_0$4

The learned operator is implemented either on full coarse fields or on local patches

$u_0$5

and produces the fine-scale value $u_0$6 at arbitrary query points (Zhang et al., 2023).

A fourth formulation is encode–approximate–reconstruct. Here the operator is decomposed as

$u_0$7

where $u_0$8 is a linear encoder defined by learned basis functions, $u_0$9 is a finite-dimensional nonlinear approximator, and $u$0 is a linear reconstructor. Multifidelity enters because the numerical encoder depends only on quadrature over the current discretization, so one model can be trained across heterogeneous resolutions without redesigning the architecture (Hauck et al., 9 Jul 2025).

These formulations share a common principle: the low-fidelity model is not discarded. It acts as a structured prior, a control variate, or an explicit input channel that reduces the complexity of the high-fidelity learning problem.

3. Architectural families

Several architectural families now define the field.

The composite multifidelity DeepONet couples three subnetworks: a low-fidelity DeepONet, a linear high-fidelity correlation subnet, and a nonlinear high-fidelity correlation subnet. The low-fidelity subnet learns the coarse operator; the linear subnet captures approximately linear cross-fidelity structure; the nonlinear subnet learns the remaining discrepancy. In the data-driven form, the high-fidelity prediction is the sum of the linear and nonlinear corrections, and in the physics-informed form the same structure is combined with PDE residual, initial-condition, and boundary-condition losses (Howard et al., 2022).

MF-WNO adapts the wavelet neural operator to multifidelity data through a two-stage architecture. First, an LF-WNO learns the low-fidelity operator on a large dataset. Then an HF-WNO is trained on high-fidelity data with input supplementation and residual operator learning, so that the final prediction is

$u$1

Because the wavelet kernel integral layer works in a multiscale representation, this design is particularly suited to localized or high-frequency inter-fidelity discrepancies (Thakur et al., 2022).

MF-LNO uses a Laplace Neural Operator as the base class and decomposes the high-fidelity correction into parallel linear and nonlinear branches. After training an LF LNO $u$2, two high-fidelity correctors are learned on the augmented input $u$3: a linear corrector $u$4 and a nonlinear corrector $u$5. The final output is

$u$6

with a learned global scalar $u$7 that balances linear and nonlinear inter-fidelity structure (Zheng et al., 1 Feb 2025).

Other architectures emphasize discretization invariance rather than explicit residual branches. GFN-ROM uses a Graph Feedforward Network to attach encoder and decoder weights to mesh nodes and transport them between meshes by nearest-neighbor rules, enabling a single reduced-order model to train on arbitrary meshes and reconstruct on unseen resolutions (Morrison et al., 2024). A more general encode–approximate–reconstruct architecture based on learned encoder and reconstructor bases extends this idea to a discretization-independent operator-learning framework whose only discretization-sensitive component is numerical quadrature (Hauck et al., 9 Jul 2025).

Family	Low-fidelity component	High-fidelity mechanism
Composite MF-DeepONet	Low-fidelity DeepONet	Linear and nonlinear correlation subnetworks
MF-WNO	LF-WNO or LF solver	Residual operator on augmented inputs
MF-LNO	LF LNO	Parallel linear/nonlinear correctors with learned $u$8
MFON closure	GPOD/PROM solver	DeepONet closure operator in reduced space
GFN-ROM / discretization-independent E-A-R	Coarse or mixed-resolution snapshots	Shared latent operator across discretizations

A recurring architectural distinction is between “composite” and “non-composite” designs. Composite models learn an explicit low-fidelity neural operator and then a high-fidelity correction. Non-composite models use a physics-based low-fidelity solver directly and learn only the discrepancy, as in the closure-corrected PROM of MFON (Ahmed et al., 2023).

4. Training strategies, physics constraints, and uncertainty quantification

Training protocols are as important as architecture. In composite multifidelity DeepONet, the data-driven objective combines a low-fidelity data loss, a high-fidelity correction loss, and regularization terms on the low-fidelity and nonlinear branches. The physics-informed variant replaces the high-fidelity data loss with PDE residual, initial-condition, and boundary-condition losses, while retaining the low-fidelity data term. This makes low-fidelity simulations act as a prior over the operator manifold while the physics terms enforce high-fidelity consistency (Howard et al., 2022).

Two-stage training is standard in MF-WNO and MF-LNO. In both cases, the low-fidelity operator is trained first on abundant low-fidelity data and then frozen. The high-fidelity stage learns a discrepancy operator on top of the frozen low-fidelity prediction, which stabilizes optimization when the high-fidelity dataset is small (Thakur et al., 2022, Zheng et al., 1 Feb 2025). This differs from mixed-fidelity pooling, which simply combines low- and high-fidelity samples into one dataset; the MF-LNO experiments explicitly compare against such a “Mix” baseline.

For time-dependent reduced models, one-step residual fitting can be insufficient. In the multifidelity DeepONet closure method, offline training feeds the network true reduced states and learns single-step residuals, but in deployment the network receives its own predicted states. To address the resulting train–test mismatch, the model is trained “in-the-loop”: the PROM and DeepONet are coupled during training, automatic differentiation passes through the PROM integrator, and the loss is imposed on trajectory windows of length $u$9 rather than a single step. This significantly improves long-horizon stability and extrapolation, although excessively large $\widetilde G_\theta^{(\mathcal S_U,\mathcal S_V)}$0 can cause gradient issues (Ahmed et al., 2023).

Bayesian variants address the fact that fine-scale labels are often scarce and noisy. The Bayesian DeepONet homogenization framework uses a factorized Gaussian likelihood,

$\widetilde G_\theta^{(\mathcal S_U,\mathcal S_V)}$1

and approximates the posterior over network parameters with SG-reLD, a stochastic gradient replica exchange Langevin diffusion sampler. Predictions are then ensemble means and variances over weight samples, yielding credible intervals and robustness to noisy fine observations (Zhang et al., 2023). MF-LNO uses a related replica-exchange SGLD strategy, reSGLD, to obtain posterior samples for uncertainty-aware prediction (Zheng et al., 1 Feb 2025).

A narrower but conceptually related line treats multifidelity approximation of expected quantities as a modified stochastic bandit. Under a linear model assumption, AETC adaptively explores multiple fidelities and then commits to the model subset with the lowest estimated conditional mean-squared error, without requiring a hierarchical model structure or prior knowledge of inter-model correlations (Xu et al., 2021). This suggests a resource-allocation perspective on operator learning, although the underlying analysis is formulated for expectation estimation rather than full function-to-function regression.

5. Applications and empirical behavior

The empirical literature spans reduced-order closure, multiscale PDE downscaling, stochastic PDE surrogates, dynamical-system operators, and inverse design.

In reduced-order closure, the multifidelity DeepONet correction for GPOD ROMs was tested on one-dimensional viscous Burgers shock advection and two-dimensional Navier–Stokes vortex merger. The closure-corrected PROM substantially improved both interpolative and extrapolative predictions, and a Pareto plot for Burgers showed that MFON improved accuracy at only $\widetilde G_\theta^{(\mathcal S_U,\mathcal S_V)}$2 additional computational cost per PROM step compared to GPOD (Ahmed et al., 2023).

In coarse-to-fine downscaling, Bayesian DeepONet homogenization maps were validated on 1D and 2D elliptic problems and on a high-contrast multiscale radiative transfer equation. Patch-based inputs consistently improved accuracy, and experiments showed good accuracy with as few as $\widetilde G_\theta^{(\mathcal S_U,\mathcal S_V)}$3–$\widetilde G_\theta^{(\mathcal S_U,\mathcal S_V)}$4 fine samples in 2D problems. In noisy settings, Bayesian B-DON reduced error relative to a noisy deterministic DON and approached the performance of DON trained on noise-free labels as the number of fine observations increased (Zhang et al., 2023).

Wavelet-based multifidelity neural operators were evaluated on artificial benchmarks, stochastic Poisson, Darcy flow in an irregular domain, the stochastic heat equation, and unsteady Allen–Cahn dynamics. Across these examples, MF-WNO achieved one to several orders of magnitude reduction in test MSE over a high-fidelity single-fidelity WNO, and in a scaling study for Darcy flow the test MSE reached by MFSM-WNO with 30 HF samples required more than 600 HF samples for HFSM-WNO (Thakur et al., 2022).

Laplace-based multifidelity neural operators were tested on the Lorenz system, Duffing oscillator, Burgers equation, and the Brusselator reaction–diffusion system. The reported testing losses were reduced by $\widetilde G_\theta^{(\mathcal S_U,\mathcal S_V)}$5 to $\widetilde G_\theta^{(\mathcal S_U,\mathcal S_V)}$6 compared to traditional approaches, even in regimes with only 10 high-fidelity samples and, for Brusselator, a single high-fidelity sample (Zheng et al., 1 Feb 2025).

Multifidelity DeepONet for PDE learning was also applied to nanoscale heat transport governed by the phonon Boltzmann transport equation. There the low-fidelity model was the same solver run with 2 source iterations and the high-fidelity model used 5 iterations. The method significantly reduced the required amount of high-fidelity data and achieved one order of magnitude smaller error when using the same amount of high-fidelity data. Once trained, the surrogate was coupled with genetic algorithms and topology optimization for fast inverse design of pore configurations (Lu et al., 2022).

Resolution-invariant reduced operator learning adds a distinct multifidelity axis. GFN-ROM was trained on mixtures of tiny, small, medium, and large meshes for Graetz flow, an advection-dominated diffusion problem, and a Stokes benchmark with seven parameters. In several cases, replacing half of the largest-mesh training data with lower-fidelity data preserved or improved the large-mesh test error, while the architecture remained substantially lighter and faster than a graph-convolutional baseline (Morrison et al., 2024). More general discretization-independent encode–approximate–reconstruct models report that multifidelity training significantly improves accuracy and computational efficiency, and that multifidelity training further enhances empirical discretization independence (Hauck et al., 9 Jul 2025).

6. Theory, misconceptions, and open problems

A central misconception is that multifidelity operator learning is inherently hierarchical and always requires a learned low-fidelity neural operator. The literature shows otherwise. Some methods are composite and learn both low- and high-fidelity operators; some are non-composite and use a physics-based low-fidelity solver directly; and some recent work frames fidelity as a property of discretization rather than of a separate model class (Ahmed et al., 2023, Hauck et al., 9 Jul 2025). A related misconception is that discretization invariance comes “for free.” Standard DeepONet is not invariant to the discretization of input functions, so patch-based multifidelity homogenization requires all patches to share the same discretization (Zhang et al., 2023).

Theoretical guarantees are unevenly distributed across the field. The discretization-independent encode–approximate–reconstruct framework establishes uniform universal approximation under strong assumptions and statistical approximation under weaker conditions, while GFN-ROM derives super-/sub-resolution error bounds controlled by mesh closeness and weight norms (Hauck et al., 9 Jul 2025, Morrison et al., 2024). By contrast, the MFON closure paper explicitly notes that it does not give formal error bounds for the multifidelity residual-learning setting or for the coupled PROM–DeepONet system, even though empirical errors approach the POD projection error in its experiments (Ahmed et al., 2023).

Another recurrent issue is dependence on low-/high-fidelity correlation. MF-WNO, MF-LNO, and multifidelity DeepONet all rely on the low-fidelity model capturing enough coarse structure that the residual is easier to learn than the full operator. When the low-fidelity model is weak or qualitatively wrong, the residual may cease to be simple, and the multifidelity benefit can erode (Thakur et al., 2022, Zheng et al., 1 Feb 2025, Lu et al., 2022). In MF-LNO this challenge is partly addressed by combining linear and nonlinear correctors, but the learned weight $\widetilde G_\theta^{(\mathcal S_U,\mathcal S_V)}$7 is global rather than input-dependent, so the paper itself notes that a fixed global weighting may be suboptimal when the correlation type changes across parameter regimes (Zheng et al., 1 Feb 2025).

Open problems therefore cluster around four themes. The first is sharper theory: extension of DeepONet and neural-operator error analysis to residual, multistage, and coupled multifidelity systems. The second is discretization robustness: designing architectures that remain accurate across irregular or changing representations without requiring common sensor sets or patch grids. The third is uncertainty and budgeting: Bayesian formulations improve robustness but add computational overhead, and bandit-style allocation methods have so far been analyzed mainly for expectation estimation rather than full operator learning (Zhang et al., 2023, Xu et al., 2021). The fourth is domain expansion. The current literature is dominated by PDE surrogates, but the policy-alignment framework ALPHA suggests that a broader notion of multifidelity operator learning can include policy operators over heterogeneous, non-hierarchical simulators, where the “operator” of interest is a high-fidelity control policy rather than a PDE solution map (Agrawal et al., 2024).

Across these directions, the unifying idea remains stable: a high-fidelity operator is rarely learned most efficiently from high-fidelity data alone. The low-fidelity model, whether it is a solver, a reduced model, a coarse discretization, a homogenized approximation, or another surrogate, is used to impose structure on the learning problem so that scarce accurate information is spent on correction rather than rediscovery.