Deep Operator Networks (DeepONets) have become a central tool in data-driven operator learning, providing flexible surrogates for nonlinear mappings arising in partial differential equations (PDEs). However, the standard trunk design based on fully connected layers acting on raw spatial or spatiotemporal coordinates struggles to represent sharp gradients, boundary layers, and non-periodic structures commonly found in PDEs posed on bounded domains with Dirichlet or Neumann boundary conditions. To address these limitations, we introduce the Spectral-Embedded DeepONet (SEDONet), a new DeepONet variant in which the trunk is driven by a fixed Chebyshev spectral dictionary rather than coordinate inputs. This non-periodic spectral embedding provides a principled inductive bias tailored to bounded domains, enabling the learned operator to capture fine-scale non-periodic features that are difficult for Fourier or MLP trunks to represent. SEDONet is evaluated on a suite of PDE benchmarks including 2D Poisson, 1D Burgers, 1D advection-diffusion, Allen-Cahn dynamics, and the Lorenz-96 chaotic system, covering elliptic, parabolic, advective, and multiscale temporal phenomena, all of which can be viewed as canonical problems in computational mechanics. Across all datasets, SEDONet consistently achieves the lowest relative L2 errors among DeepONet, FEDONet, and SEDONet, with average improvements of about 30-40% over the baseline DeepONet and meaningful gains over Fourier-embedded variants on non-periodic geometries. Spectral analyses further show that SEDONet more accurately preserves high-frequency and boundary-localized features, demonstrating the value of Chebyshev embeddings in non-periodic operator learning. The proposed architecture offers a simple, parameter-neutral modification to DeepONets, delivering a robust and efficient spectral framework for surrogate modeling of PDEs on bounded domains.

Environment Conservation Ocean Underwater robot survives voyage to'never-accessed region of the planet' An Argo float delivered unprecedented data after eight months beneath Antarctic ice. Breakthroughs, discoveries, and DIY tips sent every weekday. Thanks to an underwater survey robot, oceanographers are getting the first-ever readings collected from underneath East Antarctic's vast ice shelves . But for a moment, it wasn't clear when(or if) the bright yellow float would return to the ocean's surface after it dove underneath the ice. "We got lucky," Steve Rintoul, an oceanographer with Australia's Commonwealth Scientific and Industrial Research Organization (CSIRO), said in a statement .

We present an explainable AI-enhanced supervisory control framework for multi-agent robotics that combines (i) a timed-automata supervisor for safe, auditable mode switching, (ii) robust continuous control (Lyapunov-based controller for large-angle maneuver; sliding-mode controller (SMC) with boundary layers for precision and disturbance rejection), and (iii) an explainable predictor that maps mission context to gains and expected performance (energy, error). Monte Carlo-driven optimization provides the training data, enabling transparent real-time trade-offs. We validated the approach in two contrasting domains, spacecraft formation flying and autonomous underwater vehicles (AUVs). Despite different environments (gravity/actuator bias vs. hydrodynamic drag/currents), both share uncertain six degrees of freedom (6-DOF) rigid-body dynamics, relative motion, and tight tracking needs, making them representative of general robotic systems. In the space mission, the supervisory logic selects parameters that meet mission criteria. In AUV leader-follower tests, the same SMC structure maintains a fixed offset under stochastic currents with bounded steady error. In spacecraft validation, the SMC controller achieved submillimeter alignment with 21.7% lower tracking error and 81.4% lower energy consumption compared to Proportional-Derivative PD controller baselines. At the same time, in AUV tests, SMC maintained bounded errors under stochastic currents. These results highlight both the portability and the interpretability of the approach for safety-critical, resource-constrained multi-agent robotics.

Graph neural networks (GNNs) have emerged as powerful surrogates for mesh-based computational fluid dynamics (CFD), but training them on high-resolution unstructured meshes with hundreds of thousands of nodes remains prohibitively expensive. We study a \emph{coarse-to-fine curriculum} that accelerates convergence by first training on very coarse meshes and then progressively introducing medium and high resolutions (up to \(3\times10^5\) nodes). Unlike multiscale GNN architectures, the model itself is unchanged; only the fidelity of the training data varies over time. We achieve comparable generalization accuracy while reducing total wall-clock time by up to 50\%. Furthermore, on datasets where our model lacks the capacity to learn the underlying physics, using curriculum learning enables it to break through plateaus.

This work proposes a Variational Physics-Informed Neural Network (VPINN) framework that integrates the Petrov-Galerkin formulation with deep neural networks (DNNs) for solving one-dimensional singularly perturbed boundary value problems (BVPs) and parabolic partial differential equations (PDEs) involving one or two small parameters. The method adopts a nonlinear approximation in which the trial space is defined by neural network functions, while the test space is constructed from hat functions. The weak formulation is constructed using localized test functions, with interface penalty terms introduced to enhance numerical stability and accurately capture boundary layers. Dirichlet boundary conditions are imposed via hard constraints, and source terms are computed using automatic differentiation. Numerical experiments on benchmark problems demonstrate the effectiveness of the proposed method, showing significantly improved accuracy in both the $L_2$ and maximum norms compared to the standard VPINN approach for one-dimensional singularly perturbed differential equations (SPDEs).

Traditional wind farm control operates each turbine independently to maximize individual power output. However, coordinated wake steering across the entire farm can substantially increase the combined wind farm energy production. Although dynamic closed-loop control has proven effective in flow control applications, wind farm optimization has relied primarily on static, low-fidelity simulators that ignore critical turbulent flow dynamics. In this work, we present the first reinforcement learning (RL) controller integrated directly with high-fidelity large-eddy simulation (LES), enabling real-time response to atmospheric turbulence through collaborative, dynamic control strategies. Our RL controller achieves a 4.30% increase in wind farm power output compared to baseline operation, nearly doubling the 2.19% gain from static optimal yaw control obtained through Bayesian optimization. These results establish dynamic flow-responsive control as a transformative approach to wind farm optimization, with direct implications for accelerating renewable energy deployment to net-zero targets.

Large language models (LLMs) have shown remarkable progress in mathematical problem-solving, but evaluation has largely focused on problems that have exact analytical solutions or involve formal proofs, often overlooking approximation-based problems ubiquitous in applied science and engineering. To fill this gap, we build on prior work and present HARDMath2, a dataset of 211 original problems covering the core topics in an introductory graduate applied math class, including boundary-layer analysis, WKB methods, asymptotic solutions of nonlinear partial differential equations, and the asymptotics of oscillatory integrals. This dataset was designed and verified by the students and instructors of a core graduate applied mathematics course at Harvard. We build the dataset through a novel collaborative environment that challenges students to write and refine difficult problems consistent with the class syllabus, peer-validate solutions, test different models, and automatically check LLM-generated solutions against their own answers and numerical ground truths. Evaluation results show that leading frontier models still struggle with many of the problems in the dataset, highlighting a gap in the mathematical reasoning skills of current LLMs. Importantly, students identified strategies to create increasingly difficult problems by interacting with the models and exploiting common failure modes. This back-and-forth with the models not only resulted in a richer and more challenging benchmark but also led to qualitative improvements in the students' understanding of the course material, which is increasingly important as we enter an age where state-of-the-art language models can solve many challenging problems across a wide domain of fields.

Current research on LoRA primarily focuses on minimizing the number of fine-tuned parameters or optimizing its architecture. However, the necessity of all fine-tuned LoRA layers during inference remains underexplored. In this paper, we investigate the contribution of each LoRA layer to the model's ability to predict the ground truth and hypothesize that lower-layer LoRA modules play a more critical role in model reasoning and understanding. To address this, we propose a simple yet effective method to enhance the performance of large language models (LLMs) fine-tuned with LoRA. Specifically, we identify a ``boundary layer'' that distinguishes essential LoRA layers by analyzing a small set of validation samples. During inference, we drop all LoRA layers beyond this boundary. We evaluate our approach on three strong baselines across four widely-used text generation datasets. Our results demonstrate consistent and significant improvements, underscoring the effectiveness of selectively retaining critical LoRA layers during inference.

We study the eigenvalues and eigenfunctions of a differential operator that governs the asymptotic behavior of the unsupervised learning algorithm known as Locally Linear Embedding when a large data set is sampled from an interval or disc. In particular, the differential operator is of second order, mixed-type, and degenerates near the boundary. We show that a natural regularity condition on the eigenfunctions imposes a consistent boundary condition and use the Frobenius method to estimate pointwise behavior. We then determine the limiting sequence of eigenvalues analytically and compare them to numerical predictions. Finally, we propose a variational framework for determining eigenvalues on other compact manifolds.

Fast-turn around methods to predict airfoil trailing-edge noise are crucial for incorporating noise limitations into design optimization loops of several applications. Among these aeroacoustic predictive models, Amiet's theory offers the best balance between accuracy and simplicity. The accuracy of the model relies heavily on precise wall-pressure spectrum predictions, which are often based on single-equation formulations with adjustable parameters. These parameters are calibrated for particular airfoils and flow conditions and consequently tend to fail when applied outside their calibration range. This paper introduces a new wall-pressure spectrum empirical model designed to enhance the robustness and accuracy of current state-of-the-art predictions while widening the range of applicability of the model to different airfoils and flow conditions. The model is developed using AI-based symbolic regression via a genetic-algorithm-based approach, and applied to a dataset of wall-pressure fluctuations measured on NACA 0008 and NACA 63018 airfoils at multiple angles of attack and inflow velocities, covering turbulent boundary layers with both adverse and favorable pressure gradients. Validation against experimental data (outside the training dataset) demonstrates the robustness of the model compared to well-accepted semi-empirical models. Finally, the model is integrated with Amiet's theory to predict the aeroacoustic noise of a full-scale wind turbine, showing good agreement with experimental measurements.