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.

Probabilistic Predictions of Process-Induced Deformation in Carbon/Epoxy Composites Using a Deep Operator Network

Fiber reinforcement and polymer matrix respond differently to manufacturing conditions due to mismatch in coefficient of thermal expansion and matrix shrinkage during curing of thermosets. These heterogeneities generate residual stresses over multiple length scales, whose partial release leads to process-induced deformation (PID), requiring accurate prediction and mitigation via optimized non-isothermal cure cycles. This study considers a unidirectional AS4 carbon fiber/amine bi-functional epoxy prepreg and models PID using a two-mechanism framework that accounts for thermal expansion/shrinkage and cure shrinkage. The model is validated against manufacturing trials to identify initial and boundary conditions, then used to generate PID responses for a diverse set of non-isothermal cure cycles (time-temperature profiles). Building on this physics-based foundation, we develop a data-driven surrogate based on Deep Operator Networks (DeepONets). A DeepONet is trained on a dataset combining high-fidelity simulations with targeted experimental measurements of PID. We extend this to a Feature-wise Linear Modulation (FiLM) DeepONet, where branch-network features are modulated by external parameters, including the initial degree of cure, enabling prediction of time histories of degree of cure, viscosity, and deformation. Because experimental data are available only at limited time instances (for example, final deformation), we use transfer learning: simulation-trained trunk and branch networks are fixed and only the final layer is updated using measured final deformation. Finally, we augment the framework with Ensemble Kalman Inversion (EKI) to quantify uncertainty under experimental conditions and to support optimization of cure schedules for reduced PID in composites.

Kinetic-Mamba: Mamba-Assisted Predictions of Stiff Chemical Kinetics

Accurate chemical kinetics modeling is essential for combustion simulations, as it governs the evolution of complex reaction pathways and thermochemical states. In this work, we introduce Kinetic-Mamba, a Mamba-based neural operator framework that integrates the expressive power of neural operators with the efficient temporal modeling capabilities of Mamba architectures. The framework comprises three complementary models: (i) a standalone Mamba model that predicts the time evolution of thermochemical state variables from given initial conditions; (ii) a constrained Mamba model that enforces mass conservation while learning the state dynamics; and (iii) a regime-informed architecture employing two standalone Mamba models to capture dynamics across temperature-dependent regimes. We additionally develop a latent Kinetic-Mamba variant that evolves dynamics in a reduced latent space and reconstructs the full state on the physical manifold. We evaluate the accuracy and robustness of Kinetic-Mamba using both time-decomposition and recursive-prediction strategies. We further assess the extrapolation capabilities of the model on varied out-of-distribution datasets. Computational experiments on Syngas and GRI-Mech 3.0 reaction mechanisms demonstrate that our framework achieves high fidelity in predicting complex kinetic behavior using only the initial conditions of the state variables.

Hybrid Iterative Solvers with Geometry-Aware Neural Preconditioners for Parametric PDEs

The convergence behavior of classical iterative solvers for parametric partial differential equations (PDEs) is often highly sensitive to the domain and specific discretization of PDEs. Previously, we introduced hybrid solvers by combining the classical solvers with neural operators for a specific geometry 1, but they tend to under-perform in geometries not encountered during training. To address this challenge, we introduce Geo-DeepONet, a geometry-aware deep operator network that incorporates domain information extracted from finite element discretizations. Geo-DeepONet enables accurate operator learning across arbitrary unstructured meshes without requiring retraining. Building on this, we develop a class of geometry-aware hybrid preconditioned iterative solvers by coupling Geo-DeepONet with traditional methods such as relaxation schemes and Krylov subspace algorithms. Through numerical experiments on parametric PDEs posed over diverse unstructured domains, we demonstrate the enhanced robustness and efficiency of the proposed hybrid solvers for multiple real-world applications.

A Neural-Operator Preconditioned Newton Method for Accelerated Nonlinear Solvers

We propose a novel neural preconditioned Newton (NP-Newton) method for solving parametric nonlinear systems of equations. To overcome the stagnation or instability of Newton iterations caused by unbalanced nonlinearities, we introduce a fixed-point neural operator (FPNO) that learns the direct mapping from the current iterate to the solution by emulating fixed-point iterations. Unlike traditional line-search or trust-region algorithms, the proposed FPNO adaptively employs negative step sizes to effectively mitigate the effects of unbalanced nonlinearities. Through numerical experiments we demonstrate the computational efficiency and robustness of the proposed NP-Newton method across multiple real-world applications, especially for very strong nonlinearities.