# Operator Learning

Training five operator models on the one-dimensional discrete Poisson problem shows that error is not learned wavenumber by wavenumber but is retained along small-eigenvalue directions of the kernel that governs the training update. Sine-aligned spectral and linear Fourier neural operators give each wavenumber its own rate, so error collects at high wavenumbers; dense networks have eigendirections close to random in the sine basis, so error is nearly equal at every wavenumber and relative error grows only because the true solution spectrum decays. Removing the minibatch cross terms leaves the GELU dense network with under 0.1% of squared error below ηλ≈8×10⁻⁵ under either forcing.

Training five operator models on the one-dimensional discrete Poisson problem shows that error is not learned wavenumber by wavenumber but is retained along small-eigenvalue directions of the kernel that governs the training update. Sine-aligned spectral and linear Fourier neural operators give each wavenumber its own rate, so error collects at high wavenumbers; dense networks have eigendirections close to random in the sine basis, so error is nearly equal at every wavenumber and relative error grows only because the true solution spectrum decays. Removing the minibatch cross terms leaves the GELU dense network with under 0.1% of squared error below ηλ≈8×10⁻⁵ under either forcing.

Rebuilding a 128×128 vorticity field from 8×8 observations is a 16× upscaling in which the coarse grid resolves only wavenumbers up to 4, so fine vortical structure must be inferred rather than interpolated. SIMR-NO splits the reconstruction into 32→64 and 64→128 stages, each adding a Fourier-layer residual correction gated by radial wavenumber on top of a bicubic base, with local convolutions recovering features beyond the truncated Fourier basis. Its mean relative error is 0.2506 on Kolmogorov-forced turbulence, below LapSRN, FNO and EDSR, and about 3% on decaying homogeneous isotropic turbulence at Re = 1000, 4000 and 8000, with little change across Reynolds number.

Rebuilding a 128×128 vorticity field from 8×8 observations is a 16× upscaling in which the coarse grid resolves only wavenumbers up to 4, so fine vortical structure must be inferred rather than interpolated. SIMR-NO splits the reconstruction into 32→64 and 64→128 stages, each adding a Fourier-layer residual correction gated by radial wavenumber on top of a bicubic base, with local convolutions recovering features beyond the truncated Fourier basis. Its mean relative error is 0.2506 on Kolmogorov-forced turbulence, below LapSRN, FNO and EDSR, and about 3% on decaying homogeneous isotropic turbulence at Re = 1000, 4000 and 8000, with little change across Reynolds number.

With phasor measurement units installed at only a few generator buses and reporting intermittently, a physics-informed graph neural operator maps a measurement window directly to the rotor-angle and speed trajectories of all machines, including those without sensors, and trains without state labels. It imposes the swing equations as a hard constraint, or as a soft one with a learned correction that absorbs model error; on the 9-bus, Kundur, 39-bus and 118-bus systems it is the most accurate estimator that neither optimizes per scenario nor uses state labels, except on the 9-bus system at large model error, where a tuned Kalman smoother is better.

With phasor measurement units installed at only a few generator buses and reporting intermittently, a physics-informed graph neural operator maps a measurement window directly to the rotor-angle and speed trajectories of all machines, including those without sensors, and trains without state labels. It imposes the swing equations as a hard constraint, or as a soft one with a learned correction that absorbs model error; on the 9-bus, Kundur, 39-bus and 118-bus systems it is the most accurate estimator that neither optimizes per scenario nor uses state labels, except on the 9-bus system at large model error, where a tuned Kalman smoother is better.

EDP-PI-DOSnet embeds classical energy-stable schemes into a physics-informed operator-splitting network so that the learned discrete operators satisfy a discrete energy dissipation law. The earlier PI-DOSnet did not guarantee that its learned linear operator was symmetric negative semidefinite, energy could rise between steps, and long-time extrapolation needed two or three rounds of retraining. On the Allen–Cahn equation, both EDP-PI-DOSnet variants keep the modified energy non-increasing at every observed time level and give smaller long-time prediction errors without retraining.

EDP-PI-DOSnet embeds classical energy-stable schemes into a physics-informed operator-splitting network so that the learned discrete operators satisfy a discrete energy dissipation law. The earlier PI-DOSnet did not guarantee that its learned linear operator was symmetric negative semidefinite, energy could rise between steps, and long-time extrapolation needed two or three rounds of retraining. On the Allen–Cahn equation, both EDP-PI-DOSnet variants keep the modified energy non-increasing at every observed time level and give smaller long-time prediction errors without retraining.

Training neural operators needs expensive numerical solution labels, while PDE inputs and governing equations are cheap physical information. PI-Init spends that information on initialization: it pretrains on unlabeled inputs with physics losses, then keeps the parameters for supervised fine-tuning. At matched label budgets, FNO prediction error falls 53.3% on Burgers and 62.7% on 1D Advection, and on Burgers 200 labeled trajectories beat supervised-only training with 2,000.

Training neural operators needs expensive numerical solution labels, while PDE inputs and governing equations are cheap physical information. PI-Init spends that information on initialization: it pretrains on unlabeled inputs with physics losses, then keeps the parameters for supervised fine-tuning. At matched label budgets, FNO prediction error falls 53.3% on Burgers and 62.7% on 1D Advection, and on Burgers 200 labeled trajectories beat supervised-only training with 2,000.

Daedalus separates function-representation learning from operator learning: a functional autoencoder is pre-trained only on analytic functions synthesized on the fly, with no solver and no labels, and the latent space it induces is frozen and reused across tasks. On 28 benchmark problems spanning fluid dynamics, solid mechanics, acoustics, and the Helmholtz and Poisson equations, operators on the frozen representation have lower relative error than the end-to-end framework GAOT on 23 of them and train 2.1 times faster per problem on average; that comparison excludes the one-time backbone pre-training.

Daedalus separates function-representation learning from operator learning: a functional autoencoder is pre-trained only on analytic functions synthesized on the fly, with no solver and no labels, and the latent space it induces is frozen and reused across tasks. On 28 benchmark problems spanning fluid dynamics, solid mechanics, acoustics, and the Helmholtz and Poisson equations, operators on the frozen representation have lower relative error than the end-to-end framework GAOT on 23 of them and train 2.1 times faster per problem on average; that comparison excludes the one-time backbone pre-training.

GLOBE compresses three-dimensional beam–field states into a compact latent space, learns the dominant evolution with a control-conditioned neural operator, and captures fine-scale residuals with a generative model, then applies parameter-free constraint operators after decoding. On two high-resolution 3D benchmarks, bunched-beam transport and plasma-wakefield acceleration, relative error is 9% and 13% lower than the strongest baseline DPOT, mid-frequency spectral error is 54% and 20% lower, and hours of particle simulation become about a minute per rollout.

GLOBE compresses three-dimensional beam–field states into a compact latent space, learns the dominant evolution with a control-conditioned neural operator, and captures fine-scale residuals with a generative model, then applies parameter-free constraint operators after decoding. On two high-resolution 3D benchmarks, bunched-beam transport and plasma-wakefield acceleration, relative error is 9% and 13% lower than the strongest baseline DPOT, mid-frequency spectral error is 54% and 20% lower, and hours of particle simulation become about a minute per rollout.