Time-Dependent Differential Equations (TDDEs) model dynamical processes across science and engineering, but time-critical applications require solvers delivering high-fidelity trajectories under stringent latency constraints. Most existing TDDE solvers are limited by time discretization, forcing a latency-accuracy trade-off where smaller step sizes capture high-fidelity trajectories but incur prohibitive runtime, while larger steps meet real-time budgets at the cost of trajectory distortion. Dynamical System Machines (DSMs) offer a promising alternative by computing through continuous-time physical evolution, yet existing DSMs struggle to capture the spatiotemporal complexity of TDDEs. This work introduces DS-TS, a novel TDDE solver that achieves both high-accuracy and ultra-efficiency, leveraging the continuous-time computation of DSMs. DS-TS integrates three key innovations: (1) Excitatory-Inhibitory Inspired Coupling to better model complex spatial interactions; (2) State-aware Dynamic Non-linearity to enable rich inter-node interactions and state-dependent spatiotemporal correlations; and (3) Hierarchical Temporal Integration to capture long-range temporal dependencies. Experiments demonstrate that DS-TS achieves high-fidelity solutions while delivering orders-of-magnitude improvements in speed ($\sim 10^3\times$) and energy efficiency ($\sim 10^5\times$) compared to baseline solvers.
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Time-Dependent Differential Equations (TDDEs) model dynamical processes across science and engineering, but time-critical applications require solvers delivering high-fidelity trajectories under stringent latency constraints. Most existing TDDE solvers are limited by time discretization, forcing a latency-accuracy trade-off where smaller step sizes capture high-fidelity trajectories but incur prohibitive runtime, while larger steps meet real-time budgets at the cost of trajectory distortion. Dynamical System Machines (DSMs) offer a promising alternative by computing through continuous-time physical evolution, yet existing DSMs struggle to capture the spatiotemporal complexity of TDDEs. This work introduces DS-TS, a novel TDDE solver that achieves both high-accuracy and ultra-efficiency, leveraging the continuous-time computation of DSMs. DS-TS integrates three key innovations: (1) Excitatory-Inhibitory Inspired Coupling to better model complex spatial interactions; (2) State-aware Dynamic Non-linearity to enable rich inter-node interactions and state-dependent spatiotemporal correlations; and (3) Hierarchical Temporal Integration to capture long-range temporal dependencies. Experiments demonstrate that DS-TS achieves high-fidelity solutions while delivering orders-of-magnitude improvements in speed ($\sim 10^3\times$) and energy efficiency ($\sim 10^5\times$) compared to baseline solvers.
Applications · Time Series
Time series prototype learning is fundamentally challenged by observational ambiguity. Discrete architectures fail to resolve this, as they lack the capacity to decouple stochastic noise from continuous dynamics. Furthermore, rigid closed-set assumptions fail to capture unseen diversity. To address these limitations, we propose a hierarchical ordinary differential equation clustering network, which utilizes neural ordinary differential equation to model latent state evolution as a continuous integral curve. This formulation enforces temporal continuity to effectively disentangle smooth feature trends from stochastic noise, while our adaptive hierarchical mechanism autonomously identifies the optimal number of prototypes without rigid prior constraints. Validated on the early link failure detection task with irregularly sampled time series, the proposed method effectively extracts underlying physical prototypes, thereby enabling robust failure detection.
Applications · Chemistry, Physics, and Earth Sciences
Recovering multiple physical parameters from high-dimensional optical measurements remains challenging in computational optics. We present *MMPD-Bench*, a pioneering benchmark that reframes multi-polarimetric modalities decomposition from Mueller matrix observations as a *modality fission* problem under the multi-modal learning paradigm. By replacing iterative numerical inversion with deep surrogate models, MMPD-Bench provides data, standardized solutions and evaluations to address the multi-physics modalities generation challenge. We benchmark representative architectures to this problem, including state-space models, vision transformers, conditional diffusion models, and neural operators, under a multi-faceted evaluation protocol that jointly assesses perceptual fidelity, physical consistency, robustness, and computational efficiency. Our analysis reveals non-trivial trade-offs between accuracy and robustness in accelerated high-fidelity polarimetric decomposition, highlighting key limitations of existing surrogates. To support reproducible research, we open-source the full codebase, together with a large-scale dataset of 21,412 high-resolution Mueller matrix observations acquired through extensive polarimetric measurements. We invite the community to further advance the intersection of polarization optics and multimodal representation learning.
Deep Learning · Attention Mechanisms
Large language models have achieved remarkable success in recent years, primarily due to self-attention. However, traditional Softmax attention suffers from numerical instability and reduced performance as the number of inference tokens increases. This work addresses these issues by proposing a new design principle for attention, viewing it as a two-stage process. The first stage (normalisation) refines standard attention by replacing Softmax with the more numerically stable Softplus followed by $l_{1}$-normalisation. Furthermore, we introduce a dynamic scale factor based on invariance entropy. We show that this novel attention mechanism outperforms conventional Softmax attention, and state-of-the-art Softmax-free alternatives. Our second proposal is to introduce a second processing stage (sharpening) which consists of a re-weighting mechanism that amplifies significant attentional weights while diminishing weaker ones. This enables the model to concentrate more effectively on relevant tokens, mitigating the attention sink phenomenon, and fundamentally improving length extrapolation. This novel, two-stage, replacement for self-attention is shown to ensure numerical stability and dramatically improve length extrapolation, maintaining a nearly constant validation loss at 16$\times$ the training length while achieving superior results on challenging long-context retrieval tasks and standard downstream benchmarks. Furthermore, symbolic regression experiments demonstrate that our method enables models to recover Newton's gravitational law from orbital trajectory sequences, providing evidence that appropriate attention mechanisms are crucial in order for foundation models to develop genuine physical world models.
Applications · Chemistry, Physics, and Earth Sciences
Autoregressive learning of time-stepping operators offers an effective approach to data-driven PDE simulation on grids. For conservation laws, however, long-horizon rollouts are often destabilized when learned updates violate global conservation and, in many applications, additional state bounds—such as nonnegative mass and densities or concentrations constrained to [0,1]. Enforcing these coupled constraints via direct next-state regression remains difficult. We introduce a framework for learning conservative transport operators on regular grids, inspired by lattice Boltzmann–style discrete-velocity transport representations. Instead of predicting the next state, the model outputs local transport operators that update cells through neighborhood exchanges, guaranteeing discrete conservation by construction. For bounded quantities, we parameterize transport within a capacity-constrained feasible set, enforcing bounds structurally rather than by post-hoc clipping. We validate FluxNet on convection-diffusion, shallow water equations, traffic flow, and spinodal decomposition. Experiments on shallow-water equations and traffic flow show improved rollout stability and physical consistency over strong baselines. On phase-field spinodal decomposition, the method enables large time-steps with long-range transport, accelerating simulation while preserving microstructure evolution in both pointwise and statistical measures.
AVGen-Bench: A Task-Driven Benchmark for Multi-Granular Evaluation of Text-to-Audio-Video Generation
Applications · Computer Vision
Text-to-Audio-Video (T2AV) generation is rapidly becoming a core interface for media creation, yet its evaluation remains fragmented. Existing benchmarks largely assess audio and video in isolation or rely on coarse embedding similarity, failing to capture fine-grained joint correctness required by realistic prompts. We introduce AVGen-Bench, a task-driven benchmark for T2AV generation, featuring high-quality prompts across 11 real-world categories. To support comprehensive assessment, we propose a multi-granular evaluation framework that combines lightweight specialist models with Multimodal Large Language Models (MLLMs), enabling evaluation from perceptual quality to fine-grained semantic controllability. Our evaluation reveals a pronounced gap between strong audio-visual aesthetics and weak semantic reliability, including persistent failures in text rendering, speech coherence, physical reasoning, and universal breakdown in musical pitch control.
Applications · Computer Vision
Existing image inpainting frameworks rely on strictly supervised training paradigms, often suffering from an over-reliance on ground-truth reconstruction, which leads to conservative outputs with misaligned creativity and limited diversity. To this end, we propose the first framework to explore Group Relative Policy Optimization (GRPO) and Direct Preference Optimization (DPO) for text-guided image inpainting, formulating an efficient online reinforcement learning pipeline that enables flexible, human-aligned aesthetic control via a preference scoring model. Crucially, by decoupling the rigid one-to-one correspondence between text prompts and masked images, our method enables the model to explore diverse, controllable, and high-quality solutions beyond a single target. Furthermore, to balance semantic consistency with physical naturalness at mask boundaries, we introduce a scale-aware dynamic reward mechanism that adaptively emphasizes boundary gradient coherence for small occlusions while prioritizing visual aesthetics in large-scale generation. Extensive experiments demonstrate that our approach consistently produces higher-quality results across different backbone architectures such as Stable Diffusion and FLUX, significantly enhancing the generative capacity of base models. Code is available at https://anonymous.4open.science/r/E3F47R.
Applications · Chemistry, Physics, and Earth Sciences
Physics-informed diffusion models typically impose PDE constraints only on the final output, leaving intermediate features unconstrained. This can enable shortcut solutions that fit training statistics yet generalize poorly under shifted boundary conditions. We introduce \textbf{REPA-P}, a \emph{teacher-free} physics-informed representation alignment framework that uses first-principles residuals as supervision. REPA-P attaches lightweight projection heads to a few early/mid layers of a diffusion backbone, decodes hidden activations into physical states, and applies PDE and boundary-condition residual losses to these intermediate predictions during training. The heads are discarded at inference, preserving the original architecture and sampling cost. Across three 2D scientific field benchmarks (Darcy flow, topology optimization, and Electrostatic Charge Potential), REPA-P accelerates convergence, reduces physics residuals by up to 80\%, and improves out-of-distribution robustness to boundary-condition shifts while maintaining generation quality with zero inference overhead. Ablations show that supervising only a small set of intermediate layers captures most gains and complements output-level physics losses.
Applications · Health / Medicine
Reliable epidemic forecasting is critical for public health decision-making yet remains challenging due to data sparsity and the non-stationary nature of disease dynamics. While recent hybrid models attempt to integrate mechanistic principles with data-driven approaches, they often relegate mechanistic priors to merely auxiliary features or regularization terms. This design not only obscures the interpretability of the mechanistic contribution but also fails to inherit the capability of physical models to generalize under non-stationary dynamics, as the core architecture remains predominantly data-driven. To address these limitations, we propose EpiDiff, a unified framework that synergizes epidemiological domain knowledge with the generative power of diffusion models. Unlike methods that rigidly fuse features, EpiDiff employs a novel uncertainty-aware steering mechanism during inference. Specifically, we quantify the posterior uncertainty of mechanistic estimations and use it to dynamically modulate the diffusion process. Extensive experiments on real-world datasets demonstrate that EpiDiff consistently outperforms state-of-the-art baselines in accuracy and robustness, particularly under non-stationary distributions, while offering transparent insights into model reliance by explicitly visualizing when the forecast is governed by mechanistic laws versus data-driven patterns. Our code and datasets are available at https://anonymous.4open.science/r/epidiff-4782.
Vafa et al. recently showed that a transformer fails to acquire an internal Newtonian world model when trained on synthetic planetary-motion data. How can we fix this problem? We find that inductive biases are key to learning the veridical world model: (1) **Spatial smoothness** is required for any world model to be learned. However, naive tokenization may disrupt smoothness since two close points in physical space may be far apart in token embedding space without sufficient training or data. We fix this by formulating the prediction problem as regression instead of classification. (2) **Spatial stability** makes the prediction robust to noise, which is not guaranteed by default, but can be taught via correcting in-context noise perturbations. (3) With both spatial smoothness and stability built in, further imposing **temporal locality** induces a Newtonian world model, while the lack of this knowledge induces a Keplerian world model -- fitting elliptical parameters instead of computing gravitational forces. Our results suggest that even simple general inductive biases are powerful enough to induce correct and specific world models. The inductive biases do not need to know that much about the underlying law to be learned, but without them, it is impossible to learn.
Social Aspects · Safety
Large Vision-Language Models (LVLMs) exhibit sophisticated reasoning but remain susceptible to object hallucination. Deviating from the prevailing attention intensity assumption, we reveal a deeper dynamic structural misalignment: hallucination is triggered at decision-critical steps where specific attention heads, acting as risky mediators, decouple from visual evidence to lock onto language priors. This establishes a pathological shortcut that bypasses visual grounding. To dismantle this, we propose Fox (Faithfulness and Observational-flow via eXpression-rectification), a training-free inference-time framework. Fox diagnoses structural misalignment using a visual attention entropy probe to localize risky mediators unsupervisedly. We then execute a targeted causal intervention via numerical logit saturation to physically sever the shortcut path. Finally, a conflict-gated cooperative decoding strategy reconciles interventional faithfulness with observational fluency. Extensive experiments demonstrate that Fox achieves SOTA performance, outperforming SID by 29.1% while preserving linguistic richness and general reasoning capabilities.
Applications · Chemistry, Physics, and Earth Sciences
Current data-driven scientific modeling struggles with a functional dichotomy: neural operators exhibit spectral bias in high-frequency regimes, while physics-constrained paradigms suffer from optimization pathologies. To bridge this gap, we propose Adaptive Physics-Informed Computing (APIC), a neuro-symbolic meta-architecture designed with structural reconfigurability to encode diverse domain priors. Crucially, APIC integrates a gradient-isolated interaction strategy that mechanistically decouples the optimization paths of parameter identification and residual correction, effectively mitigating gradient interference/conflicts. By instantiating this framework for nonlinear dissipative systems, we derive the Generalized Kuramoto-Sivashinsky-Cahn-Hilliard (G-KSCH) kernel, providing a unified representation for sparse dynamic identification. Extensive experiments demonstrate that APIC establishes new benchmarks in 3D compressible supersonic shock wave prediction, surpassing diverse architectures (e.g., CNNs and Transformers) by 20% to 70% in predictive accuracy. Notably, APIC achieves Pareto-optimal performance, delivering superior precision with reduced computational overhead compared to SOTA models, while exhibiting robust cross-domain generalization across meteorological and urban traffic datasets.
General Machine Learning · Kernel methods
Kernel methods are powerful tools in statistical learning, but their cubic complexity in the sample size $n$ limits their use on large-scale datasets. In this work, we introduce a scalable framework for kernel regression with complexity $O(n \log n)$, fully leveraging GPU acceleration. The approach is based on a Fourier representation of kernels combined with non-uniform fast Fourier transforms (NUFFT), enabling exact, fast, and memory-efficient computations. We instantiate our framework in three settings: Sobolev kernel regression, physics-informed regression, and additive models. When known, the proposed estimators are shown to achieve minimax convergence rates, consistent with classical kernel theory. Empirical results demonstrate that our methods can process up to tens of billions of samples within minutes, providing both statistical accuracy and computational scalability. These contributions establish a flexible approach, paving the way for the routine application of kernel methods in large-scale learning tasks, whenever the kernel norm can be efficiently expressed in the Fourier space and the ambient dimension $d$ is small.
Applications · Robotics
Vision-language models demonstrate strong reasoning and planning abilities, yet grounding these predictions into precise robot actions remains a central challenge. Existing Vision-Language-Action methods typically entangle reasoning and action generation, leading to limited generalization and costly adaptation. We propose to learn a \textbf{G}eneralizable \textbf{A}ction \textbf{E}xpert (\textbf{GAE}), a task-agnostic model that converts sparse geometric plans into dense robot actions. Our approach introduces a sparse geometric interface: the VLM predicts sparse 3D waypoints representing high-level intention, while GAE maps these waypoints together with real-time point cloud observations to continuous action trajectories. GAE is pretrained on a large-scale pointcloud–trajectory dataset comprising \textbf{150k} trajectories from both simulation and real-world robots. To further improve efficiency and generalization, we introduce an \textbf{Action Pre-training, Pointcloud Fine-tuning (APPF)} scheme that decouples learning action dynamics from geometry grounding. After pretraining, GAE is frozen and reused across downstream tasks, requiring only lightweight fine-tuning of the VLM to produce the sparse interface. Extensive experiments show that our method achieves strong performance and generalization across diverse visual domains, camera viewpoints, and natural language instructions.
Applications · Chemistry, Physics, and Earth Sciences
DNA language models are increasingly used to represent genomic sequence, yet their effectiveness depends critically on how raw nucleotides are converted into model inputs. Unlike natural language, DNA offers no canonical “word” boundaries, making fixed tokenizations a brittle design choice under shifts, indels, and local repeats. We introduce DNAChunker, a masked DNA language model that incorporates a learnable adaptive segmentation module to produce context-dependent, variable-length units. Building on a dynamic segmentation procedure, DNAChunker learns to allocate finer granularity to functionally enriched regions while compressing repetitive or redundant sequence. We pre-train DNAChunker on the human reference genome (HG38) and evaluate it on the Nucleotide Transformer and Genomic Benchmarks, where it consistently improves over strong fixed-tokenization baselines. Further analyses and ablations indicate that the learned segmentation is structured rather than incidental: the model preferentially uses shorter units around promoters and exons, and longer units in repetitive regions, yielding representations that are both mutation-resilient and biologically-informed.
Theory · Optimization
Physics-Informed Neural Networks (PINNs) often train slowly or fail to converge on challenging partial differential equations (PDEs), a behavior recently linked to severely ill-conditioned loss landscapes inherited from the underlying differential operator. We propose FK-PINNs, a simple modification of the PINN objective that provably improves this conditioning: at a few points in the domain we compute Feynman--Kac estimates of the solution by Monte Carlo averaging, and add the resulting data-fidelity term to the standard residual and boundary losses. For a broad class of linear second-order PDEs admitting a Feynman--Kac representation, we show that this term acts as an operator-level preconditioner: for suitable weights, our comparison bounds guarantee a substantially smaller condition number than under the standard PINN loss, even for modest Monte Carlo sample budgets. Leveraging learning-theoretic tools, we derive non-asymptotic $L^2(\Omega)$-error bounds for the FK-PINNs with $\tanh$ activation by decomposing the excess risk into approximation, statistical, and optimization error terms and tightly controlling the Monte Carlo error tails. Along the way, we establish pseudo-dimension bounds for first- and second-order derivatives of $\tanh$ neural networks, which are of independent interest and, to the best of our knowledge, new. Numerical experiments on Poisson, Schrödinger, mean exit time, and committor problems corroborate the theory, and show that FK-PINNs can successfully solve PDEs for which vanilla PINNs exhibit severe failure modes.
Applications · Computer Vision
Predicting 3D geometric trajectory requires capturing complex spatiotemporal dependencies while preserving physical symmetries. While flow matching offers a powerful generative paradigm, extending it to SE(3)-equivariant dynamics is challenging due to the inherent gap between deterministic history and stochastic evolving flows. To address this, we introduce GSE-Flow, an SE(3)-equivariant flow matching framework. We first propose a Coherent Sequence Encoding and Time-Modulated Embedding strategy that unifies historical and evolving streams, incorporating velocity and flow time via equivariant affine transformations to guide continuous evolution. We further design a Geometry-Feature Tensorization mechanism that projects node states into a tensor product space, enabling Context-Flow Fusion to guide trajectory evolution with historical context. GSE-Flow guarantees theoretical SE(3)-equivariance and achieves SOTA accuracy on MD17, MD22, and CMU MoCap benchmarks for geometric trajectory prediction, while demonstrating generality by enhancing deterministic baselines.
Applications · Chemistry, Physics, and Earth Sciences
Quantum hardware suffers from intrinsic device heterogeneity and environmental drift, forcing practitioners to choose between suboptimal non-adaptive controllers or costly per-device recalibration. We derive a scaling law lower bound for meta-learning showing that the adaptation gain (expected fidelity improvement from task-specific gradient steps) saturates exponentially with gradient steps and scales linearly with task variance, providing a quantitative criterion for when adaptation justifies its overhead. Validation on quantum gate calibration shows negligible benefits for low-variance tasks but $>40$% fidelity gains on two-qubit gates under extreme out-of-distribution conditions (10$\times$ the training noise), with implications for reducing per-device calibration time on cloud quantum processors. Further validation on classical linear-quadratic control confirms these laws emerge from general optimization geometry rather than quantum-specific physics. Together, these results offer a transferable framework for decision-making in adaptive control.
Applications · Chemistry, Physics, and Earth Sciences
This work addresses the problem of learning the dynamics of high-dimensional probability densities over time using unlabeled samples, without assuming access to trajectory information. We introduce two-parameter flows that learn only sampling-time transports from a base distribution to each marginal and then extract a physics-time velocity by regressing on coupled synthetic trajectories. We prove that the resulting physics-time dynamics are unique and inherit regularity from the sampling-time transports. Because we can build on standard, well-developed conditional flow matching techniques for learning the base-to-marginal transports, our approach scales to high dimensions and avoids per-step optimal-transport couplings, while allowing admissible non-gradient dynamics that can naturally explain rotational or circulating physics phenomena.