Equation Recast for Canonical Operator Learning Across Parametric PDEs

Learning solution operators across broad parameter ranges can require substantial coverage of both input functions and physical parameters, particularly for purely data-driven parametric models. In addition, the resulting models may fail silently outside the training distribution. We introduce equation recast, which reformulates parametric operator learning as the learning of a single canonical operator. Parameter-induced operator variations are derived analytically from the governing equation and absorbed into effective sources, enabling zero-shot prediction across new parameter regimes. Across multi-parameter, nonlinear, and singular PDE settings, equation recast supports extrapolation, integrates sparse heterogeneous datasets in a shared canonical representation, and uses loss of convergence as an internal warning signal for failure of the recast iteration. In high-fidelity tokamak simulations for nuclear fusion, the framework unifies electron-temperature data across four device geometries through canonical-domain mapping within one jointly trained operator. Equation recast provides a route toward reusable neural PDE solvers combining equation-guided transfer, data efficiency, and monitorable inference.

Conformal risk control for model-form uncertainty in parametric non-intrusive reduced-order models

Non-intrusive reduced-order models (NIROMs) have become a standard tool for approximating parametric partial differential equations from computer design of experiments while significantly reducing computational costs. However, assessing the reliability of their predictions remains a major challenge, particularly in extrapolation regimes or under limited training data. In this work, we introduce a framework for quantifying model-form uncertainty in NIROMs by combining a perturbative stochastic representation of reduced bases with distribution-free conformal-type methods. Starting from a deterministic reduced basis constructed from snapshot matrices, we model uncertainty through random perturbations defined on the Stiefel manifold, directed along the discarded modes, yielding stochastic reduced-order approximations whose induced variance reflects the basis-truncation error. A transport approximation gives a closed-form posterior variance that sepa- rates basis-induced from regression-induced uncertainty, without re-training the underlying Gaussian processes. We include this posterior variance within a conformal risk control calibration framework, that provides prediction sets with coordinate miscoverage guarantees. The calibration factor produced by this framework is itself an interpretable, scalar diagnostic of the quality of the uncertainty estimate. The methodology is evaluated on parametric PDE benchmarks and an industrial tire-manufacturing calendering process. Numerical experiments demonstrate reliable, locally informative uncertainty quantification that goes beyond the Gaussian predictive variance.

Randomized neural operator for parametric PDEs with fast training and conformal uncertainty quantification

Repeatedly solving parametric PDEs is essential for uncertainty quantification, design optimization and inverse problems, but conventional neural operators require expensive non-convex training. We introduce PCA--RaNN, a randomized latent neural operator that combines PCA-based dimensionality reduction with fixed random features and a closed-form least-squares readout. It recasts latent operator learning as fixed-feature linear regression, reducing training time by one to three orders of magnitude across benchmarks while maintaining competitive accuracy. We introduce an energy-matched scaling rule and a lightweight two-parameter BFGS refinement to correct suboptimal feature scales. Ensemble averaging reduces predictive variance. On Burgers, Darcy, Navier--Stokes and backward heat equation benchmarks, PCA--RaNN provides a favorable speed--accuracy trade-off against operator-learning baselines. The ensemble supports split-conformal prediction intervals, and the linear readout enables rapid online adaptation via recursive least squares without retraining hidden features. This provides an efficient, uncertainty-aware surrogate for many-query scientific workflows.

Loss-Conditional PINNs for Parametric PDE Families

Physics-informed neural networks (PINNs) approximate solutions of ODEs and PDEs by minimising a weighted combination of residual, boundary, initial, and data losses. Their performance is often dominated by the choice of loss weights: a poor weighting can drive training to a degenerate solution in which one physical constraint is satisfied while another is ignored. Existing methods select or adapt a single good set of weights. We take a different view: instead of tuning one weight vector, we explore the entire weight space during training. We introduce LC-PINN, which adapts the loss-conditional training of Dosovitskiy and Djolonga (2020) to the PDE-residual setting: the conditioning vector (either the loss weights or a scalar physical coefficient) is treated as a network input and sampled from a simple prior at every optimisation step. This turns PINN training into learning a continuous family of solutions indexed by that vector, with no solver-generated paired data. LC-PINN thus lies between classical PINNs and operator learning: it stays fully physics-informed but amortises training over a parametric family. Our contribution is not the loss-conditional construction itself, but its extension to PINNs, the unification of the loss-weight and parametric-coefficient regimes under one architecture (concatenation for loss weights, FiLM for coefficients), and a fixed-quadrature L-BFGS finishing protocol that makes the parametric-coefficient regime trainable. We give a lambda-invariance result for the conditional optimum and study LC-PINN on parametric Helmholtz, Schrodinger, viscous Burgers, and Buckley-Leverett equations. A single LC-PINN matches or improves retrained per-weight PINN baselines while parameterising the full family in one model, at a total cost that amortises favourably against per-instance retraining.

Shearlet Neural Operators for Anisotropic-Shock-Dominated and Multi-scale parametric partial differential equations

Neural operators have emerged as powerful data-driven surrogates for learning solution operators of parametric partial differential equations (PDEs). However, widely used Fourier Neural Operators (FNOs) rely on global Fourier representations, which can be inefficient for resolving anisotropic structures, sharp gradients, and spatially localized discontinuities that arise in shock-dominated and multiscale regimes. To address these limitations, we introduce the Shearlet Neural Operator (SNO), a neural operator architecture that replaces the Fourier transform with a shearlet-based representation. Shearlets offer directional, multiscale, and spatially localized atoms with near-optimal sparse approximation of anisotropic features, providing an inductive bias aligned with PDE solutions containing edges, fronts, and shocks. SNO learns in the shearlet domain and reconstructs predictions via the inverse transform, retaining efficient spectral computation while improving locality and directional selectivity. Across seven benchmark PDE families, including strongly anisotropic advection, anisotropic diffusion, and nonlinear conservation laws with straight, curved, interacting, spiral, and polygonal shock structures, SNO consistently improves predictive accuracy and feature fidelity over FNO baselines, with the largest gains observed in anisotropic and discontinuity-dominated settings.