Physics-Guided Dimension Reduction for Simulation-Free Operator Learning of Stiff Differential--Algebraic Systems

Neural surrogates for stiff differential-algebraic equations (DAEs) face two key challenges: soft-constraint methods leave algebraic residuals that stiffness amplifies into large errors, while hard-constraint methods require trajectory data from computationally expensive stiff integrators. We introduce an extended Newton implicit layer that enforces algebraic consistency and quasi-steady-state reduction within a single differentiable solve. Given slow-state predictions from a physics-informed DeepONet, the proposed layer recovers fast and algebraic states, eliminates the stiffness-amplification pathway within each time window, and reduces the output dimension to the slow states alone. Gradients derived via the implicit function theorem capture a stiffness-scaled coupling term that is absent in penalty-based approaches. Cascaded implicit layers further extend the framework to multi-component systems with provable convergence. On a grid-forming inverter DAE (21 states), the proposed method (7 outputs, 1.42 percent error) significantly outperforms penalty methods (39.3 percent), standard Newton approaches (57.0 percent), and augmented Lagrangian or feedback linearization baselines, which fail to converge. Two independently trained models compose into a 44-state system without retraining, achieving 0.72 to 1.16 percent error with zero algebraic residual. Conformal prediction further provides 90 percent coverage in-distribution and enables automatic out-of-distribution detection.

Jeffreys Flow: Robust Boltzmann Generators for Rare Event Sampling via Parallel Tempering Distillation

Sampling physical systems with rough energy landscapes is hindered by rare events and metastable trapping. While Boltzmann generators already offer a solution, their reliance on the reverse Kullback--Leibler divergence frequently induces catastrophic mode collapse, missing specific modes in multi-modal distributions. Here, we introduce the Jeffreys Flow, a robust generative framework that mitigates this failure by distilling empirical sampling data from Parallel Tempering trajectories using the symmetric Jeffreys divergence. This formulation effectively balances local target-seeking precision with global modes coverage. We show that minimizing Jeffreys divergence suppresses mode collapse and structurally corrects inherent inaccuracies via distillation of the empirical reference data. We demonstrate the framework's scalability and accuracy on highly non-convex multidimensional benchmarks, including the systematic correction of stochastic gradient biases in Replica Exchange Stochastic Gradient Langevin Dynamics and the massive acceleration of exact importance sampling in Path Integral Monte Carlo for quantum thermal states.

AIVV: Neuro-Symbolic LLM Agent-Integrated Verification and Validation for Trustworthy Autonomous Systems

Deep learning models excel at detecting anomaly patterns in normal data. However, they do not provide a direct solution for anomaly classification and scalability across diverse control systems, frequently failing to distinguish genuine faults from nuisance faults caused by noise or the control system's large transient response. Consequently, because algorithmic fault validation remains unscalable, full Verification and Validation (V\&V) operations are still managed by Human-in-the-Loop (HITL) analysis, resulting in an unsustainable manual workload. To automate this essential oversight, we propose Agent-Integrated Verification and Validation (AIVV), a hybrid framework that deploys Large Language Models (LLMs) as a deliberative outer loop. Because rigorous system verification strictly depends on accurate validation, AIVV escalates mathematically flagged anomalies to a role-specialized LLM council. The council agents perform collaborative validation by semantically validating nuisance and true failures based on natural-language (NL) requirements to secure a high-fidelity system-verification baseline. Building on this foundation, the council then performs system verification by assessing post-fault responses against NL operational tolerances, ultimately generating actionable V\&V artifacts, such as gain-tuning proposals. Experiments on a time-series simulator for Unmanned Underwater Vehicles (UUVs) demonstrate that AIVV successfully digitizes the HITL V\&V process, overcoming the limitations of rule-based fault classification and offering a scalable blueprint for LLM-mediated oversight in time-series data domains.

pADAM: A Plug-and-Play All-in-One Diffusion Architecture for Multi-Physics Learning

Generalizing across disparate physical laws remains a fundamental challenge for artificial intelligence in science. Existing deep-learning solvers are largely confined to single-equation settings, limiting transfer across physical regimes and inference tasks. Here we introduce pADAM, a unified generative framework that learns a shared probabilistic prior across heterogeneous partial differential equation families. Through a learned joint distribution of system states and, where applicable, physical parameters, pADAM supports forward prediction and inverse inference within a single architecture without retraining. Across benchmarks ranging from scalar diffusion to nonlinear Navier--Stokes equations, pADAM achieves accurate inference even under sparse observations. Combined with conformal prediction, it also provides reliable uncertainty quantification with coverage guarantees. In addition, pADAM performs probabilistic model selection from only two sparse snapshots, identifying governing laws through its learned generative representation. These results highlight the potential of generative multi-physics modeling for unified and uncertainty-aware scientific inference.

Muon with Spectral Guidance: Efficient Optimization for Scientific Machine Learning

Physics-informed neural networks and neural operators often suffer from severe optimization difficulties caused by ill-conditioned gradients, multi-scale spectral behavior, and stiffness induced by physical constraints. Recently, the Muon optimizer has shown promise by performing orthogonalized updates in the singular-vector basis of the gradient, thereby improving geometric conditioning. However, its unit-singular-value updates may lead to overly aggressive steps and lack explicit stability guarantees when applied to physics-informed learning. In this work, we propose SpecMuon, a spectral-aware optimizer that integrates Muon's orthogonalized geometry with a mode-wise relaxed scalar auxiliary variable (RSAV) mechanism. By decomposing matrix-valued gradients into singular modes and applying RSAV updates individually along dominant spectral directions, SpecMuon adaptively regulates step sizes according to the global loss energy while preserving Muon's scale-balancing properties. This formulation interprets optimization as a multi-mode gradient flow and enables principled control of stiff spectral components. We establish rigorous theoretical properties of SpecMuon, including a modified energy dissipation law, positivity and boundedness of auxiliary variables, and global convergence with a linear rate under the Polyak-Lojasiewicz condition. Numerical experiments on physics-informed neural networks, DeepONets, and fractional PINN-DeepONets demonstrate that SpecMuon achieves faster convergence and improved stability compared with Adam, AdamW, and the original Muon optimizer on benchmark problems such as the one-dimensional Burgers equation and fractional partial differential equations.

Structured Unitary Tensor Network Representations for Circuit-Efficient Quantum Data Encoding

Encoding classical data into quantum states is a central bottleneck in quantum machine learning: many widely used encodings are circuit-inefficient, requiring deep circuits and substantial quantum resources, which limits scalability on quantum hardware. In this work, we propose TNQE, a circuit-efficient quantum data encoding framework built on structured unitary tensor network (TN) representations. TNQE first represents each classical input via a TN decomposition and then compiles the resulting tensor cores into an encoding circuit through two complementary core-to-circuit strategies. To make this compilation trainable while respecting the unitary nature of quantum operations, we introduce a unitary-aware constraint that parameterizes TN cores as learnable block unitaries, enabling them to be directly optimized and directly encoded as quantum operators. The proposed TNQE framework enables explicit control over circuit depth and qubit resources, allowing the construction of shallow, resource-efficient circuits. Across a range of benchmarks, TNQE achieves encoding circuits as shallow as $0.04\times$ the depth of amplitude encoding, while naturally scaling to high-resolution images ($256 \times 256$) and demonstrating practical feasibility on real quantum hardware.

Neural-POD: A Plug-and-Play Neural Operator Framework for Infinite-Dimensional Functional Nonlinear Proper Orthogonal Decomposition

The rapid development of AI for Science is often hindered by the "discretization", where learned representations remain restricted to the specific grids or resolutions used during training. We propose the Neural Proper Orthogonal Decomposition (Neural-POD), a plug-and-play neural operator framework that constructs nonlinear, orthogonal basis functions in infinite-dimensional space using neural networks. Unlike the classical Proper Orthogonal Decomposition (POD), which is limited to linear subspace approximations obtained through singular value decomposition (SVD), Neural-POD formulates basis construction as a sequence of residual minimization problems solved through neural network training. Each basis function is obtained by learning to represent the remaining structure in the data, following a process analogous to Gram--Schmidt orthogonalization. This neural formulation introduces several key advantages over classical POD: it enables optimization in arbitrary norms (e.g., $L^2$, $L^1$), learns mappings between infinite-dimensional function spaces that is resolution-invariant, generalizes effectively to unseen parameter regimes, and inherently captures nonlinear structures in complex spatiotemporal systems. The resulting basis functions are interpretable, reusable, and enabling integration into both reduced order modeling (ROM) and operator learning frameworks such as deep operator learning (DeepONet). We demonstrate the robustness of Neural-POD with different complex spatiotemporal systems, including the Burgers' and Navier-Stokes equations. We further show that Neural-POD serves as a high performance, plug-and-play bridge between classical Galerkin projection and operator learning that enables consistent integration with both projection-based reduced order models and DeepONet frameworks.

$f$-GRPO and Beyond: Divergence-Based Reinforcement Learning Algorithms for General LLM Alignment

Recent research shows that Preference Alignment (PA) objectives act as divergence estimators between aligned (chosen) and unaligned (rejected) response distributions. In this work, we extend this divergence-based perspective to general alignment settings, such as reinforcement learning with verifiable rewards (RLVR), where only environmental rewards are available. Within this unified framework, we propose $f$-Group Relative Policy Optimization ($f$-GRPO), a class of on-policy reinforcement learning, and $f$-Hybrid Alignment Loss ($f$-HAL), a hybrid on/off policy objectives, for general LLM alignment based on variational representation of $f$-divergences. We provide theoretical guarantees that these classes of objectives improve the average reward after alignment. Empirically, we validate our framework on both RLVR (Math Reasoning) and PA tasks (Safety Alignment), demonstrating superior performance and flexibility compared to current methods.

Ultra Fast PDE Solving via Physics Guided Few-step Diffusion

Diffusion-based models have demonstrated impressive accuracy and generalization in solving partial differential equations (PDEs). However, they still face significant limitations, such as high sampling costs and insufficient physical consistency, stemming from their many-step iterative sampling mechanism and lack of explicit physics constraints. To address these issues, we propose Phys-Instruct, a novel physics-guided distillation framework which not only (1) compresses a pre-trained diffusion PDE solver into a few-step generator via matching generator and prior diffusion distributions to enable rapid sampling, but also (2) enhances the physics consistency by explicitly injecting PDE knowledge through a PDE distillation guidance. Physic-Instruct is built upon a solid theoretical foundation, leading to a practical physics-constrained training objective that admits tractable gradients. Across five PDE benchmarks, Phys-Instruct achieves orders-of-magnitude faster inference while reducing PDE error by more than 8 times compared to state-of-the-art diffusion baselines. Moreover, the resulting unconditional student model functions as a compact prior, enabling efficient and physically consistent inference for various downstream conditional tasks. Our results indicate that Phys-Instruct is a novel, effective, and efficient framework for ultra-fast PDE solving powered by deep generative models.

ATLAS : Adaptive Self-Evolutionary Research Agent with Task-Distributed Multi-LLM Supporters

Recent multi-LLM agent systems perform well in prompt optimization and automated problem-solving, but many either keep the solver frozen after fine-tuning or rely on a static preference-optimization loop, which becomes intractable for long-horizon tasks. We propose ATLAS (Adaptive Task-distributed Learning for Agentic Self-evolution), a task-distributed framework that iteratively develops a lightweight research agent while delegating complementary roles to specialized supporter agents for exploration, hyperparameter tuning, and reference policy management. Our core algorithm, Evolving Direct Preference Optimization (EvoDPO), adaptively updates the phase-indexed reference policy. We provide a theoretical regret analysis for a preference-based contextual bandit under concept drift. In addition, experiments were conducted on non-stationary linear contextual bandits and scientific machine learning (SciML) loss reweighting for the 1D Burgers' equation. Both results show that ATLAS improves stability and performance over a static single-agent baseline.