Spectral Audit of In-Context Operator Networks

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.

Operator Learning for Reconstructing Flow Fields from Sparse Measurements: a Language Model Approach

Reconstructing flow fields from sparse measurements is a fundamental problem in fluid mechanics with broad implications for modeling, control, and design. In this work, we propose a novel operator learning framework that leverages the architecture of language models to perform flow reconstruction in a mesh-free manner. We reformulate flow field reconstruction as a sequence-to-sequence learning task, where sparse measurements are treated as context and unobserved locations as queries. Our model learns to reconstruct the full flow field from sparse inputs, effectively capturing spatial correlations and long-range dependencies. We evaluate the proposed approach on four benchmark datasets: (1) two-dimensional vortex street simulations, (2) daily average temperature data across the contiguous United States, (3) three-dimensional blood flow simulations based on dissipative particle dynamics, and (4) three-dimensional turbulent jet flow measurements obtained via particle tracking velocimetry. Across all cases, our method demonstrates competitive reconstruction accuracy, even with highly incomplete data (less than 10\% observed), and achieves efficient performance. The results highlight the potential of language models as robust and scalable tools for scientific data reconstruction, and suggest a promising direction toward the development of foundation models for scientific and engineering applications.

UFO: A Domain-Unification-Free Operator Framework for Generalized Operator Learning

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.

Multi-fidelity surrogates for mechanics of composites: from co-kriging to multi-fidelity neural networks

Composite materials exhibit strongly hierarchical and anisotropic properties governed by coupled mechanisms spanning constituents, plies, laminates, structures, and manufacturing history. This intrinsic complexity makes predictive modeling of composites expensive, because repeated experiments and high-fidelity simulations are needed to cover large design spaces of material, structure, and manufacturing. Multi-fidelity surrogate modeling addresses this challenge by combining abundant, less expensive data with limited high-accuracy data to recover reliable high-fidelity predictions. This review presents a structured overview of multi-fidelity modeling for composite mechanics, covering Gaussian-process or Kriging-based methods, including co-Kriging, coregionalization models, autoregressive formulations, nonlinear autoregressive Gaussian processes, multi-fidelity deep Gaussian processes, and multi-fidelity neural networks. Their distinctions are examined in terms of cross-fidelity correlation, discrepancy representation, uncertainty quantification, and scalability. Selected examples of their applications to composites are introduced according to the roles that multi-fidelity surrogates play in engineering problems, including forward prediction for rapid exploration of material design spaces, inverse optimization for composite parameter identification and design search under limited high-fidelity access, and workflow integration, where heterogeneous data sources, constraints, and validation requirements determine model utility. Open question discussions highlight recurring challenges specific to composites, such as regime-dependent fidelity gaps associated with nonlinear damage and manufacturing history, mismatches between simulations and experiments, and uncertainty propagation across multi-fidelity models.

PINNs in PDE Constrained Optimal Control Problems: Direct vs Indirect Methods

We study physics-informed neural networks (PINNs) as numerical tools for the optimal control of semilinear partial differential equations. We first recall the classical direct and indirect viewpoints for optimal control of PDEs, and then present two PINN formulations: a direct formulation based on minimizing the objective under the state constraint, and an indirect formulation based on the first-order optimality system. For a class of semilinear parabolic equations, we derive the state equation, the adjoint equation, and the stationarity condition in a form consistent with continuous-time Pontryagin-type optimality conditions. We then specialize the framework to an Allen-Cahn control problem and compare three numerical approaches: (i) a discretize-then-optimize adjoint method, (ii) a direct PINN, and (iii) an indirect PINN. Numerical results show that the PINN parameterization has an implicit regularizing effect, in the sense that it tends to produce smoother control profiles. They also indicate that the indirect PINN more faithfully preserves the PDE contraint and optimality structure and yields a more accurate neural approximation than the direct PINN.

Curvature-Aware Optimization for High-Accuracy Physics-Informed Neural Networks

Efficient and robust optimization is essential for neural networks, enabling scientific machine learning models to converge rapidly to very high accuracy -- faithfully capturing complex physical behavior governed by differential equations. In this work, we present advanced optimization strategies to accelerate the convergence of physics-informed neural networks (PINNs) for challenging partial (PDEs) and ordinary differential equations (ODEs). Specifically, we provide efficient implementations of the Natural Gradient (NG) optimizer, Self-Scaling BFGS and Broyden optimizers, and demonstrate their performance on problems including the Helmholtz equation, Stokes flow, inviscid Burgers equation, Euler equations for high-speed flows, and stiff ODEs arising in pharmacokinetics and pharmacodynamics. Beyond optimizer development, we also propose new PINN-based methods for solving the inviscid Burgers and Euler equations, and compare the resulting solutions against high-order numerical methods to provide a rigorous and fair assessment. Finally, we address the challenge of scaling these quasi-Newton optimizers for batched training, enabling efficient and scalable solutions for large data-driven problems.

Spectral bias in physics-informed and operator learning: Analysis and mitigation guidelines

Solving partial differential equations (PDEs) by neural networks as well as Kolmogorov-Arnold Networks (KANs), including physics-informed neural networks (PINNs), physics-informed KANs (PIKANs), and neural operators, are known to exhibit spectral bias, whereby low-frequency components of the solution are learned significantly faster than high-frequency modes. While spectral bias is often treated as an intrinsic representational limitation of neural architectures, its interaction with optimization dynamics and physics-based loss formulations remains poorly understood. In this work, we provide a systematic investigation of spectral bias in physics-informed and operator learning frameworks, with emphasis on the coupled roles of network architecture, activation functions, loss design, and optimization strategy. We quantify spectral bias through frequency-resolved error metrics, Barron-norm diagnostics, and higher-order statistical moments, enabling a unified analysis across elliptic, hyperbolic, and dispersive PDEs. Through diverse benchmark problems, including the Korteweg-de Vries, wave and steady-state diffusion-reaction equations, turbulent flow reconstruction, and earthquake dynamics, we demonstrate that spectral bias is not simply representational but fundamentally dynamical. In particular, second-order optimization methods substantially alter the spectral learning order, enabling earlier and more accurate recovery of high-frequency modes for all PDE types. For neural operators, we further show that spectral bias is dependent on the neural operator architecture and can also be effectively mitigated through spectral-aware loss formulations without increasing the inference cost.

Physics-Informed Laplace Neural Operator for Solving Partial Differential Equations

Neural operators have emerged as fast surrogate solvers for parametric partial differential equations (PDEs). However, purely data-driven models often require extensive training data and can generalize poorly, especially in small-data regimes and under unseen (out-of-distribution) input functions that are not represented in the training data. To address these limitations, we propose the Physics-Informed Laplace Neural Operator (PILNO), which enhances the Laplace Neural Operator (LNO) by embedding governing physics into training through PDE, boundary condition, and initial condition residuals. To improve expressivity, we first introduce an Advanced LNO (ALNO) backbone that retains a pole-residue transient representation while replacing the steady-state branch with an FNO-style Fourier multiplier. To make physics-informed training both data-efficient and robust, PILNO further leverages (i) virtual inputs: an unlabeled ensemble of input functions spanning a broad spectral range that provides abundant physics-only supervision and explicitly targets out-of-distribution (OOD) regimes; and (ii) temporal-causality weighting: a time-decaying reweighting of the physics residual that prioritizes early-time dynamics and stabilizes optimization for time-dependent PDEs. Across four representative benchmarks -- Burgers' equation, Darcy flow, a reaction-diffusion system, and a forced KdV equation -- PILNO consistently improves accuracy in small-data settings (e.g., N_train <= 27), reduces run-to-run variability across random seeds, and achieves stronger OOD generalization than purely data-driven baselines.

An AI-enabled tool for quantifying overlapping red blood cell sickling dynamics in microfluidic assays

Understanding sickle cell dynamics requires accurate identification of morphological transitions under diverse biophysical conditions, particularly in densely packed and overlapping cell populations. Here, we present an automated deep learning framework that integrates AI-assisted annotation, segmentation, classification, and instance counting to quantify red blood cell (RBC) populations across varying density regimes in time-lapse microscopy data. Experimental images were annotated using the Roboflow platform to generate labeled dataset for training an nnU-Net segmentation model. The trained network enables prediction of the temporal evolution of the sickle cell fraction, while a watershed algorithm resolves overlapping cells to enhance quantification accuracy. Despite requiring only a limited amount of labeled data for training, the framework achieves high segmentation performance, effectively addressing challenges associated with scarce manual annotations and cell overlap. By quantitatively tracking dynamic changes in RBC morphology, this approach can more than double the experimental throughput via densely packed cell suspensions, capture drug-dependent sickling behavior, and reveal distinct mechanobiological signatures of cellular morphological evolution. Overall, this AI-driven framework establishes a scalable and reproducible computational platform for investigating cellular biomechanics and assessing therapeutic efficacy in microphysiological systems.

GIMLET: Generalizable and Interpretable Model Learning through Embedded Thermodynamics

We develop a data-driven framework for discovering constitutive relations in models of fluid flow and scalar transport. Our approach infers unknown closure terms in the governing equations (gray-box discovery) under the assumption that the temporal derivative, convective transport, and pressure-gradient contributions are known. The formulation is rooted in a variational principle from nonequilibrium thermodynamics, where the dynamics is defined by a free-energy functional and a dissipation functional. The unknown constitutive terms arise as functional derivatives of these functionals with respect to the state variables. To enable a flexible and structured model discovery, the free-energy and dissipation functionals are parameterized using neural networks, while their functional derivatives are obtained via automatic differentiation. This construction enforces thermodynamic consistency by design, ensuring monotonic decay of the total free energy and non-negative entropy production. The resulting method, termed GIMLET (Generalizable and Interpretable Model Learning through Embedded Thermodynamics), avoids reliance on a predefined library of candidate functions, unlike sparse regression or symbolic identification approaches. The learned models are generalizable in that functionals identified from one dataset can be transferred to distinct datasets governed by the same underlying equations. Moreover, the inferred free-energy and dissipation functions provide direct physical interpretability of the learned dynamics. The framework is demonstrated on several benchmark systems, including the viscous Burgers equation, the Kuramoto--Sivashinsky equation, and the incompressible Navier--Stokes equations for both Newtonian and non-Newtonian fluids.