Learning Informative Prior with Infinite-Dimensional Continuous Normalizing Flow for Bayesian Inverse Problem

This paper addresses infinite-dimensional Bayesian inference for inverse problem of partial differential equations with model parameters in infinite-dimensional Hilbert space. To effectively incorporate prior information, we propose a novel continuous normalizing flows based infinite-dimensional model. Specifically, by introducing a well-defined neural ordinary differential equation in infinite-dimensional space, a simple reference measure can be transformed into a more complex measure which encodes the prior information. A corresponding theoretical framework is established to ensure the well-posedness of our proposed Bayesian prior in infinite-dimensional space. We also provide training methods of the prior for two distinct data settings, along with two sampling algorithms for the resulting Bayesian posterior. The proposed framework is applied to three representative inverse problems: the simple smooth inverse problem, inverse scattering problem, and the inverse heat conduction problem. Numerical experiments support the theoretical analysis and demonstrate the efficiency of the proposed algorithms.

Enforcing Dirichlet Boundary Conditions in Operator Learning

Operator learning in scientific machine learning is concerned with approximation of maps between infinite-dimensional function spaces; such maps frequently arise as the solution operators of partial differential equations (PDEs). Neural operators have demonstrated broad empirical success at approximating such maps from data. However, most existing neural operator architectures enforce boundary conditions indirectly through training from data even though the boundary condition is often known exactly. Furthermore, existing modifications and approaches that do enforce boundary conditions explicitly suffer from impractical restrictions, including boundary smoothness, uniform grids, and separable, box-like domains. In this work, we propose an architecture which, independently of training, satisfies homogeneous Dirichlet boundary conditions, whilst simultaneously retaining the expressivity of existing kernel-integral neural operator architectures. This is achieved by enforcing the property that the output of each layer is contained in the span of a subset of the homogeneous Dirichlet eigenfunctions of the Laplacian on the output domain. The method requires only that the output domain be bounded with Lipschitz boundary and places no restriction on the choice of discretization, making it applicable to arbitrary mesh data and general geometries. We prove universal approximation for the resulting architecture; furthermore the approach we adopt in the analysis proves universality for a broad class of kernel-integral neural operators thereby uniting existing theory for a variety of operator learning methods. We validate the proposed method on maps defined by the coefficient to solution map in 2D PDEs: Darcy flow on a square domain and the Helmholtz equation on a circular domain. Comparisons are made with alternative methods.

Advanced Linear Algebra with Applications - Part I (Numerical linear algebra for PDEs, machine learning, and data assimilation)

These lecture notes form the first part of a master's-level course on advanced numerical linear algebra. Their aim is not only to present the classical algorithms, but to show why the subject has become considerably more central than it was a generation ago. Numerical linear algebra grew up alongside the numerical solution of partial differential equations, and for a long time that is where its large sparse systems came from. Ranking the nodes of a network, assimilating observations into a weather forecast, and fitting a model to a large noisy data set now lead to problems of the same kind: too large to factorise, structured, and accessible only through matrix-vector products. Strikingly few ideas are needed for all of them. Each chapter therefore develops a standard topic and then puts it to work outside its original setting. We treat norms, factorisations, conditioning and floating-point arithmetic; sparse matrices arising from finite differences, from graphs and from machine learning; stationary iterations and the smoothing property; the conjugate gradient and Lanczos methods, with spectral clustering and regularisation by early stopping; Arnoldi and GMRES, with PageRank and large least squares; and finally preconditioning, Schwarz domain decomposition and multigrid. We assume a first course in linear algebra. Every section closes with a summary of what should be retained and every chapter with exercises, several drawn from past examinations. Accompanying Python code reproduces the numerical illustrations.

Multi-stage neural operator learning with application for convolutions

Convolution integrals widely exist in applications, and to enable fast and accurate computations, this paper introduces two general multi-stage neural operator learning frameworks. The first, Deep Collocation Neural Operator (DCNO), is a supervised approach that iteratively refines the operator approximation by learning residuals from input-output data pairs. The second, Deep Galerkin Neural Operator (DGNO), is an unsupervised framework applicable when the target operator can be represented by a PDE, leveraging the weak form of the PDE residual for training. Both methods progressively construct basis operators through multiple training stages to enrich the approximation space, leading to significantly improved accuracy over standard one-shot operator learning. We provide theoretical analysis for their approximation capabilities and implement them for learning convolutions. Extensive numerical experiments demonstrate that both DCNO and DGNO achieve high accuracy, approaching machine precision under single float for convolution problems, and offer substantial efficiency gains for numerous queries or parametric variations compared to traditional solvers. We also extend these frameworks to handle multi-input operator learning scenarios involving variations in both the density and kernel of a convolution.

FAST-DeepONet: Factor-Augmented Branch Representations for High-Dimensional PDE Inputs in the Small-Sample Regime

Deep operator networks can become statistically unstable when partial differential equation inputs are observed at thousands of strongly correlated sensors but only a small number of operator samples is available. We introduce FAST-DeepONet, a branch representation combining a fixed spectral path with a regularized projection of the orthogonal residual, in which the directional penalty acts on the effective residual map after each of its rows is normalized. On Navier--Stokes flow a plain DeepONet degrades from $0.0394$ to $0.1556$ mean relative $L_2$ error as the branch grows from $129$ to $8193$ coordinates, while FAST-DeepONet stays near $0.04$, so the sensor grid can be refined without a statistical penalty. Across independent test sets for Navier--Stokes flow, Darcy flow, and signed terminal wavefield prediction it lowers mean relative $L_2$ error by $4.7\%$ to $37.0\%$ with three to seven times fewer trainable parameters. A spectral-only branch sharing the same basis separates the two paths: the fixed spectral path carries the improvement on Navier--Stokes and Darcy, while terminal wave prediction requires the residual path together with its directional penalty. FAST-DeepONet targets coordinate-query architectures and trains on solution values alone.

Beyond Field Accuracy: Two-Axis Diagnosis of Inverse-PINN Parameter Error

Inverse physics-informed neural networks (PINNs) can reconstruct a field accurately while returning an incorrect physical parameter. We introduce a two-axis post-training diagnosis that separates finite-sample resolution under a specified observation-and-estimation protocol from the signed parameter preference encoded by the final learned field and residual metric. The first axis repeatedly fits noisy observations with a matched forward estimator. At known synthetic truth, the second freezes the field and residual view and computes a local score displacement toward a nearby residual-profile minimum. Endpoint consistency then tests whether joint training delivers that preference under the same final view. Across three synthetic one-dimensional, scalar-parameter PDEs, matched-forward mean absolute relative error ranges from 2.34 percent to 17.46 percent. The displacement tracks frozen-profile minima across locked seeds, architectures, and fresh-noise retraining (r from .945 to .982), and it tracks delivered signed log-error in 240 fresh-noise RBA runs (r = .994; 237/240 correct directions). A coupled two-parameter Darcy check validates the full matrix calculation. The axes are complementary diagnostic coordinates, not additive error components or a deployable oracle-free estimator. Together, they route follow-up work toward observations, residual evidence, or endpoint delivery.

MiNO: Cotangent-bundle propagator learning for PDEs

Scientific machine learning for partial differential equations commonly targets solution fields, as in physics-informed neural networks, or solution maps, as in neural operators. We study a third target: the propagator itself, a phase and amplitude in phase space. The motivation is a gap in regularity. A transported discontinuity is nonsmooth in space and time, yet the rule that moves it can be a polynomial phase carrying unit amplitude, so the object that generates an evolution can be far smoother than the field it generates. The microlocal neural operator (MiNO) learns that object, using the eikonal equation for the phase and the transport equation for the amplitude, and recovers the solution by an oscillatory integral. Sharp fronts and caustics then belong to propagation geometry rather than to a field fitted pointwise. Small residuals certify more than the reconstructed field. They place the learned canonical relation, the geometry that carries singularities, close to the exact one, and they separate trainable error from the frequency-truncation tail. On a matched-budget discontinuous-advection benchmark, MiNO stops improving within 10,000 steps at the accuracy limit of its finite reconstruction window, a limit predicted in closed form, whereas a physics-informed neural network with neural-tangent-kernel loss balancing stays near its initial error. On smooth advection, the mean error is $3.84\times10^{-3}$ for MiNO and $3.12\times10^{-2}$ for a supervised Fourier neural operator. Single-branch MiNO is the smallest model compared, and one trained generator serves five unseen initial conditions without retraining.

Kernel Methods for Learning Operators with Multiple Inputs and Outputs

Learning mappings between infinite-dimensional objects is a central challenge in scientific machine learning. We introduce a general kernel-based encoder-decoder framework for operator learning that separates observation, representation, learning, and reconstruction. We develop this framework for multi-input, multi-output operator learning, where operators map between products of potentially distinct function spaces. Our approximation theory shows that, although the number of inputs and outputs can increase, the convergence rate is governed by the most challenging constituent approximation problem rather than the overall problem dimension. The framework leads to practical kernel methods with closed-form training and inference, combining mathematical tractability with computational efficiency. We further specialize the approach to multiple operator learning by introducing KernelMO, a family of kernel methods with complementary operator-valued and product-space formulations. Across five families of parametric partial differential equations, the proposed methods achieve competitive or state-of-the-art predictive accuracy while reducing training and inference costs relative to neural operator architectures and deep learning based models, offering an efficient and lightweight alternative.

Target-Guided Selective Reweighting for Physics-Informed Neural Network Inverse Problems: A Transfer Learning Approach

Physics-informed neural networks (PINNs) encounter ill-posed optimization, loss competition, and parameter compensation in partial differential equation (PDE) inverse problems. Transfer learning can reuse representations from source tasks, but direct fine-tuning may introduce negative transfer when dominant physical mechanisms, governing parameters, or observation noise differ between source and target domains: the model achieves low field error yet recovers incorrect target physical parameters. To mitigate, we propose Target-Guided Selective Reweighting PINN (TGSR-PINN), a target-evidence-driven representation correction method for PINN inverse transfer learning. TGSR-PINN transfers only the weights and biases from the source PINN, while target physical parameters are independently initialized; after a short target-adaptation phase, the method computes neuron target scores using first-order Taylor sensitivity and pre-activation variance on fixed scoring batches, and converts evidence associated with low-scoring neurons into continuous weak-adaptation signals via a Gaussian mixture model (GMM) with rank fallback. TGSR-PINN then applies selective soft decay to input weight rows and biases of low-scoring neurons instead of hard pruning or random resetting. In experiments, TGSR-PINN improves target parameter recovery while maintaining comparable field accuracy in the high-Péclet 2D advection-diffusion task and in the Allen--Cahn to Burgers cross-PDE-family transfer task; a 5%-noise reaction--diffusion case provides supplementary evidence under milder source-target mismatch. Ablation studies suggest that neuron target scoring, weak-adaptation signal estimation, layer protection, and selective soft decay jointly contribute to the benefits.

GAIA: Geometry-Adaptive Operator Learning for Forward and Inverse Problems

Operator learning for partial differential equations (PDEs) on arbitrary geometries builds fast neural surrogates for large-scale simulation. Although recent geometry-adaptive neural operators have made substantial progress, they are mainly designed for forward problems in which inputs and outputs share the same spatial domain. This limits their applicability for boundary value problems (BVPs) and inverse problems, where inputs and outputs may live on different domains. We introduce the Geometry-Adaptive Integral Autoencoder (GAIA), an operator learning model that encodes the domain boundary and the interior field distribution into geometry tokens, and conditions integral transform layers on these tokens via cross-attention, allowing the kernel to adapt locally to geometric features. This yields a single architecture for forward (including BVPs) and inverse problems on arbitrary domains in one pass, without retraining, iterative optimization, or graph construction. We evaluate GAIA on seven 2D and 3D benchmarks, four of which are new or substantially extended benchmarks for inverse problems and BVP: electrical impedance tomography, optical tomography, 3D Darcy flow on varying geometries, and a modified setting of Poisson BVP on mechanical components benchmark (MCB). GAIA sets new state-of-the-art results on every inverse and BVP task, reducing median relative $L^2$ error by 64% on airfoil flow reconstruction and 27% on EIT relative to the next best amortized method, and outperforming all baselines on every shape category of MCB. On other forward problems, GAIA is competitive with specialized solvers while maintaining stable accuracy across point resolutions on which transformer-based baselines degrade.

Convolutional Symmetric AutoEncoders: enhancing latent stability via differential geometry

Autoencoders (AEs) have emerged as powerful tools for non-linear dimensionality reduction, often surpassing traditional linear methods such as Proper Orthogonal Decomposition (POD) in scenarios characterized by slowly decaying Kolmogorov $n$-widths. In the realm of Reduced-Order Modelling (ROM), these models are increasingly utilized to learn low-dimensional representations of solution manifolds associated with parametric Partial Differential Equations (PDEs). However, the high expressivity of AEs presents a challenge: although trained networks typically minimize reconstruction error, they often struggle to capture the essential properties necessary for building accurate and robust ROMs. Recent works by arXiv:2307.15288v2 and arXiv:2506.11641v1 have tackled this challenge in fully connected AEs by proposing representation-consistent architectures, which preserve some of the properties belonging to POD. This study builds upon that concept by extending representation consistency for convolutional layers. We introduce a novel class of symmetric Convolutional AutoEncoders (CAEs) designed to embody the primary properties of manifold parametrization mappings. When integrated into a ROM framework, this architecture demonstrates significantly improved predictive capabilities. Specifically, we compared the performance of the ROMs based on classical and symmetric CAEs on three one dimensional academic test cases, namely the Linear Advection, the Viscous Burger and the Kuramoto Sivashinsky equation. Numerical results demonstrate that our proposed symmetric approach consistently yields more accurate latent trajectories, lower reconstruction errors, and enhanced model robustness.

From Spectral Methods to Sample Complexity Bounds for Fourier Neural Operators

We establish approximation and learning guarantees for Fourier neural operators (FNOs) applied to time-$T$ solution operators of dissipative evolution equations. The analysis builds on the premise that FNOs can efficiently approximate and learn solution operators whenever these operators admit stable and accurate spectral discretizations. To formalize this idea, we introduce classes of evolution operators defined through spectral methods and derive FNO approximation bounds and polynomial sample complexity guarantees for these classes. For equations with polynomial nonlinearities, the learning rates depend primarily on the smoothness of the input space and the dimension of the physical domain. Our results hold uniformly over broad families of dissipative equations, rather than for a single fixed PDE, and apply in particular to the Navier--Stokes, Allen--Cahn, and Cahn--Hilliard equations. For equations with non-polynomial smooth nonlinearities, we prove that polynomial sample complexity still holds with rates that now additionally depend on the smoothness of the nonlinear terms and the dissipation strength. Overall, we connect classical spectral approximation theory with modern operator learning and explain when FNOs can learn nonlinear evolution operators efficiently.

Fourier Neural Operators with Least-Squares Readout Refit for Learning Random Obstacle-to-Solution Maps

We study operator learning for random obstacle-to-solution maps arising from elliptic variational inequalities with finite-band self-affine random obstacle fields. Instead of introducing an explicit truncated stochastic parametrization of the random input, we learn the map directly from sampled obstacle realizations on a fixed grid. This problem is challenging because the solution is governed not only by the obstacle field itself, but also by the induced contact set and free-boundary geometry. We introduce a post-training least-squares readout refit for the Fourier neural operator (FNO). After the FNO is trained end to end, its nonlinear backbone is frozen and the final affine readout is recomputed by solving the induced linear least-squares problem over all training samples and grid points. The refit yields the empirical squared-error optimal readout for the learned frozen features while leaving the nonlinear representation unchanged. We compare vanilla DeepONet, POD-DeepONet, a two-stage DeepONet baseline, FNO, and FNO with least-squares readout refit (FNO-LS) on two obstacle ensembles with different amplitude levels. Numerical results show that FNO-LS achieves the strongest overall performance among the tested models, particularly for higher-amplitude obstacles with more complex contact geometry. The method improves average field accuracy, contact-set recovery, and obstacle-violation metrics at low additional cost, especially when the FNO backbone is informative but not fully converged. These results suggest that least-squares readout refit is a simple and effective post-training enhancement for learning random obstacle-to-solution maps.

Hessian-augmented Supervised Learning for Hamilton-Jacobi-Bellman PDEs

A data-driven method is developed for approximating value functions in deterministic optimal control problems with nonlinear control-affine dynamics. The Pontryagin Maximum Principle optimality system is solved from multiple initial conditions to generate training data consisting of values, gradients, and Hessians of the value function, where Hessian information is obtained from a matrix Riccati equation along optimal trajectories. These quantities augment a weighted least-squares regression over sparse polynomial bases on hyperbolic cross index sets, with gradients and Hessians contributing additional linear equations per sample and substantially reducing sample complexity compared to value-only regression. Feedback laws are recovered analytically from the learned value function. In high dimensions, a partial Hessian strategy controls the cost of data generation. The approach is validated on problems of increasing state dimension, where second-order data augmentation is shown to improve approximation accuracy and closed-loop performance, with up to an order-of-magnitude reduction in the number of training samples required relative to lower-order methods.

Overlapping Schwarz Attention: Hierarchical Attention via Domain Decomposition

We propose a hierarchical attention mechanism based on two-level overlapping Schwarz domain decomposition. The method is motivated by domain decomposition methods in partial differential equations which combine local subdomain corrections with a coarse level that communicates global, long-range information. We test its usefulness in the context of finite-dimensional operator learning using a simple, one-dimensional diffusion problem. Although elementary, this problem provides a controlled sequence-to-sequence setting in which the exact nonlocal solution operator is known. After discretization, learning the solution operator amounts to approximating the inverse of a symmetric positive definite matrix. As a baseline, we use a global softmax-free low-rank attention operator of the form \(QK^T\). The proposed construction replaces this global factorization by a two-level additive structure: local low-rank attention blocks on overlapping subdomains are combined with a coarse attention block. The resulting operator has the form $$ M_θ^{-1} = ΦQ_0 K_0^T Φ^T + \sum_{i=1}^{N} R_i^T D_i^{1/2} Q_i K_i^T D_i^{1/2} R_i . $$ Here, $R_i$ restricts to an overlapping subdomain, $D_i$ is a partition-of-unity weight, and $Φ$ is a coarse interpolation %(or prolongation) matrix. Numerical experiments for synthetic Fourier right-hand sides indicate that the domain-decomposition attention operator can converge faster and can give more accurate approximations than a global low-rank attention baseline while using significantly fewer parameters.

Adjoint Method versus Physics-Informed Neural Networks in PDE-Constrained Inverse Problems

Inverse problems governed by partial differential equations (PDEs) are central to computational mechanics and are commonly solved by adjoint-based optimization, while physics-informed neural networks (PINNs) have emerged as a flexible alternative. Their relative performance remains difficult to assess because the two approaches are often compared under different formulations, parameterizations, optimizers, and regularization choices. We present a fair comparison of adjoint optimization and PINNs for PDE-constrained inverse problems. From a common abstract formulation, we instantiate both methods on identical domains, governing equations, observation models, and regularization terms, while matching the optimizer, unknown parameterization, and arithmetic precision wherever applicable. The benchmarks include unsteady Burgers, noisy Darcy permeability inversion, three-dimensional Allen--Cahn reaction identification, and unsteady Navier--Stokes viscosity identification. The results show that the representation of the unknown largely determines the preferred method: grid-based fields favor the discrete adjoint, whereas neural representations are native to PINNs and relevant for closure and constitutive modeling. For time-dependent problems, adjoint inversion can be dominated by trajectory storage and differentiation, while PINNs provide satisfactory reconstructions at lower cost. A PINN-warm-started adjoint strategy then recovers adjoint-level accuracy at substantially reduced cost.