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Applications · Physics

Giacomo Baldan, Qiang Liu, Alberto Guardone, Nils Thuerey

Physics-constrained generative modeling aims to produce high-dimensional samples that are both physically consistent and distributionally accurate, a task that remains challenging due to often conflicting optimization objectives. Recent advances in flow matching and diffusion models have enabled efficient generative modeling, but integrating physical constraints often degrades generative fidelity or requires costly inference-time corrections. Our work is the first to recognize the trade-off between distributional and physical accuracy. Based on the insight of inherently conflicting objectives, we introduce Physics-Based Flow Matching (PBFM) a method that enforces physical constraints at training time using conflict-free gradient updates and unrolling to mitigate Jensen's gap. Our approach avoids manual loss balancing and enables simultaneous optimization of generative and physical objectives. As a consequence, physics constraints do not impede inference performance. We benchmark our method across three representative PDE benchmarks. PBFM achieves a Pareto-optimal trade-off, competitive inference speed, and generalizes to a wide range of physics-constrained generative tasks, providing a practical tool for scientific machine learning. Code and datasets available at [https://github.com/tum-pbs/PBFM](https://github.com/tum-pbs/PBFM).

Applications · Language, Speech and Dialog

Min-Sang Baek, Gyeong-Su Kim, Donghyun Kim, Joon-Hyuk Chang

Sound source localization (SSL) is a fundamental task in spatial audio understanding, yet most deep neural network-based methods are constrained by fixed array geometries and predefined directional grids, limiting generalizability and scalability. To address these issues, we propose _audio-geometry-grid representation learning_ (AGG-RL), a novel framework that jointly learns audio-geometry and grid representations in a shared latent space, enabling both geometry-invariant and grid-flexible SSL. Moreover, to enhance generalizability and interpretability, we introduce two physics-informed components: a _learnable non-uniform discrete Fourier transform_ (LNuDFT), which optimizes the dense allocation of frequency bins in a non-uniform manner to emphasize informative phase regions, and a _relative microphone positional encoding_ (rMPE), which encodes relative microphone coordinates in accordance with the nature of inter-channel time differences. Experiments on synthetic and real datasets demonstrate that AGG-RL achieves superior performance, particularly under unseen conditions. The results highlight the potential of representation learning with physics-informed design towards a universal solution for spatial acoustic scene understanding across diverse scenarios.

Applications · Robotics

Sunwoo Kim, Jeongjun Kim, Joseph Lim

Achieving both speed and accuracy is a central challenge for real-world robot manipulation. While recent imitation learning approaches, including vision-language-action (VLA) models, have achieved remarkable precision and generalization, their execution speed is often limited by slow demonstration via teleoperation and by inference latency. In this work, we introduce a method to accelerate any imitation policy that predicts action chunks, enabling speeds that surpass those of the original demonstration. A naive approach of simply increasing the execution frequency of predicted actions leads to significant state errors and task failure, as it alters the underlying transition dynamics and encounters physical reachability constraints over shorter time horizons. These errors are further amplified by misaligned actions based on outdated robot state when using asynchronous inference to accelerate execution. Our method $\textbf{\textit{RACE}}$ address these challenges with a three-part solution: 1) using desired states as imitation targets instead of commanded actions, 2) replanning the timing of action chunks to execute them as fast as the robot's physical limits allow, and 3) employing a test-time search for an aligned action chunk that maximizes controllability from the current state. Through extensive experiments in both simulation and the real world, we show that our method achieves up to a 4x acceleration over the original policy while maintaining a high success rate

Deep Learning · Theory

Alessio Giorlandino, Sebastian Goldt

Finding the right initialisation for neural networks is crucial to ensure smooth training and good performance. In transformers, the wrong initialisation can lead to one of two failure modes of self-attention layers: rank collapse, where all tokens collapse into similar representations, and entropy collapse, where highly concentrated attention scores lead to training instability. While previous work has studied different scaling regimes for transformers, an asymptotically exact, down-to-the constant prescription for how to initialise transformers has so far been lacking. Here, we provide an analytical theory of signal propagation through deep transformers with self-attention, layer normalisation, skip connections and MLP. Our theory yields a simple algorithm to compute trainability diagrams that identify the correct choice of initialisation hyper-parameters for a given architecture. We overcome the key challenge, an exact treatment of the self-attention layer, by establishing a formal parallel with the Random Energy Model from statistical physics. We also analyse gradients in the backward path and determine the regime where gradients vanish at initialisation. We demonstrate the versatility of our framework through three case studies. Our theoretical framework gives a unified perspective on the two failure modes of self-attention and gives quantitative predictions on the scale of both weights and residual connections that guarantee smooth training.

Deep Learning · Theory

O Duranthon, Pierre Marion, Claire Boyer, Bruno Loureiro, Lenka Zdeborova

Large language models rely on attention mechanisms with a softmax activation. Yet the dominance of softmax over alternatives (e.g., component-wise or linear) remains poorly understood, and many theoretical works have focused on the easier-to-analyze linearized attention. In this work, we address this gap through a principled study of the single-location regression task, where the output depends on a linear transformation of a single input token at a random location. Building on ideas from statistical physics, we develop an analysis of attention-based predictors in the high-dimensional limit, where generalization performance is captured by a small set of order parameters. At the population level, we show that softmax achieves the Bayes risk, whereas linear attention fundamentally falls short. We then examine other activation functions to identify which properties are necessary for optimal performance. Finally, we analyze the finite-sample regime: we provide an asymptotic characterization of the test error and show that, while softmax is no longer Bayes-optimal, it consistently outperforms linear attention. We discuss the connection with optimization by gradient-based algorithms.

Applications · Physics

Peiyan Hu, Haodong Feng, Hongyuan Liu, Tongtong Yan, Wenhao Deng, Tianrun Gao, Rong Zheng, Haoren Zheng, Chenglei Yu, Chuanrui Wang 等

Predicting the evolution of complex physical systems remains a central problem in science and engineering. Despite rapid progress in scientific Machine Learning (ML) models, a critical bottleneck is the lack of expensive real-world data, resulting in most current models being trained and validated on simulated data. Beyond limiting the development and evaluation of scientific ML, this gap also hinders research into essential tasks such as sim-to-real transfer. We introduce RealPDEBench, the first benchmark for scientific ML that integrates real-world measurements with paired numerical simulations. RealPDEBench consists of five datasets, three tasks, nine metrics, and ten baselines. We first present five real-world measured datasets with paired simulated datasets across different complex physical systems. We further define three tasks, which allow comparisons between real-world and simulated data, and facilitate the development of methods to bridge the two. Moreover, we design nine evaluation metrics, spanning data-oriented and physics-oriented metrics, and finally benchmark ten representative baselines, including state-of-the-art models, pretrained PDE foundation models, and a traditional method. Experiments reveal significant discrepancies between simulated and real-world data, while showing that pretraining with simulated data consistently improves both accuracy and convergence. In this work, we hope to provide insights from real-world data, advancing scientific ML toward bridging the sim-to-real gap and real-world deployment. Our benchmark, datasets, and instructions are available at https://realpdebench.github.io/.

Applications · Physics

David Zenati, Eliya Nachmani

Quantum Error Correction (QEC) decoding faces a fundamental accuracy-efficiency tradeoff. Classical methods like Minimum Weight Perfect Matching (MWPM) exhibit variable performance across noise models and suffer from polynomial complexity, while tensor network decoders achieve high accuracy but at prohibitively high computational cost. Recent neural decoders reduce complexity but lack the accuracy needed to compete with computationally expensive classical methods. We introduce SAQ-Decoder, a unified framework combining transformer-based learning with constraint aware post-processing that achieves both near Maximum Likelihood (ML) accuracy and linear computational scalability with respect to the syndrome size. Our approach combines a dual-stream transformer architecture that processes syndromes and logical information with asymmetric attention patterns, and a novel differentiable logical loss that directly optimizes Logical Error Rates (LER) through smooth approximations over finite fields. SAQ-Decoder achieves high accuracy decoding, with error thresholds of 10.99\% (independent noise) and 18.6\% (depolarizing noise) on toric codes that closely approach the theoretical ML bounds of 11.0\% and 18.9\% while outperforming existing neural and classical baselines in accuracy, complexity, and parameter efficiency. Our findings establish that learned decoders can simultaneously achieve competitive decoding accuracy and computational efficiency, addressing key requirements for practical fault-tolerant quantum computing systems.

Deep Learning · Algorithms

yulun wu, Miguel Aguiar, Karl H. Johansson, Matthieu Barreau

Spectral bias, the tendency of neural networks to learn low-frequency features first, is a well-known issue with many training algorithms for physics-informed neural networks (PINNs). To overcome this issue, we propose IFeF-PINN, an algorithm for iterative training of PINNs with Fourier-enhanced features. The key idea is to enrich the latent space using high-frequency components through Random Fourier Features. This creates a two-stage training problem: (i) estimate a basis in the feature space, and (ii) perform regression to determine the coefficients of the enhanced basis functions. For an underlying linear model, it is shown that the latter problem is convex, and we prove that the iterative training scheme converges. Furthermore, we empirically establish that Random Fourier Features enhance the expressive capacity of the network, enabling accurate approximation of high-frequency PDEs. Through extensive numerical evaluation on classical benchmark problems, the superior performance of our method over state-of-the-art algorithms is shown, and the improved approximation across the frequency domain is illustrated.

Applications · Time Series

Roussel Nzoyem, Nawid Keshtmand, Enrique Crespo Fernandez, Idriss TSAYEM, Raul Santos-Rodriguez, David Barton, Tom Deakin

We introduce WARP (**W**eight-space **A**daptive **R**ecurrent **P**rediction), a simple yet powerful model that unifies weight-space learning with linear recurrence to redefine sequence modeling. Unlike conventional recurrent neural networks (RNNs) which collapse temporal dynamics into fixed-dimensional hidden states, WARP explicitly parametrizes its hidden state as the weights and biases of a distinct auxiliary neural network, and uses input differences to drive its recurrence. This brain-inspired formulation enables efficient gradient-free adaptation of the auxiliary network at test-time, in-context learning abilities, and seamless integration of domain-specific physical priors. Empirical validation shows that WARP matches or surpasses state-of-the-art baselines on diverse classification tasks, featuring in the top three in 4 out of 6 real-world challenging datasets. Furthermore, extensive experiments across sequential image completion, multivariate time series forecasting, and dynamical system reconstruction demonstrate its expressiveness and generalization capabilities. Remarkably, a physics-informed variant of our model outperforms the next best model by more than 10x. Ablation studies confirm the architectural necessity of key components, solidifying weight-space linear RNNs as a transformative paradigm for adaptive machine intelligence.

Computer Vision · 3D Rendering & Reconstruction

Qianfan Shen, Ningxiao Tao, Qiyu Dai, Tianle Chen, Minghan Qin, Yongjie Zhang, Mengyu Chu, Wenzheng Chen, Baoquan Chen

We consider the problem of synthesizing photorealistic, physically plausible combustion effects in in-the-wild 3D scenes. Traditional CFD and graphics pipelines can produce realistic fire effects but rely on handcrafted geometry, expert-tuned parameters, and labor-intensive workflows, limiting their scalability to the real world. Recent scene modeling advances like 3D Gaussian Splatting (3DGS) enable high-fidelity real-world scene reconstruction, yet lack physical grounding for combustion. To bridge this gap, we propose FieryGS, a physically-based framework that integrates physically-accurate and user-controllable combustion simulation and rendering within the 3DGS pipeline, enabling realistic fire synthesis for real scenes. Our approach tightly couples three key modules: (1) multimodal large-language-model-based physical material reasoning, (2) efficient volumetric combustion simulation, and (3) a unified renderer for fire and 3DGS. By unifying reconstruction, physical reasoning, simulation, and rendering, FieryGS removes manual tuning and automatically generates realistic, controllable fire dynamics consistent with scene geometry and materials. Our framework supports complex combustion phenomena—including flame propagation, smoke dispersion, and surface carbonization—with precise user control over fire intensity, airflow, ignition location and other combustion parameters. Evaluated on diverse indoor and outdoor scenes, FieryGS outperforms all comparative baselines in visual realism, physical fidelity, and controllability.

Applications · Climate

Rafael Pablos Sarabia, Joachim Nyborg, Morten Birk, Jeppe Sjørup, Anders Vesterholt, Ira Assent

We present a deep learning model for high-resolution probabilistic precipitation forecasting over an 8-hour horizon in Europe, overcoming the limitations of radar-only deep learning models with short forecast lead times. Our model efficiently integrates multiple data sources - including radar, satellite, and physics-based numerical weather prediction (NWP) - while capturing long-range interactions, resulting in accurate forecasts with robust uncertainty quantification through consistent probabilistic maps. Featuring a compact architecture, it enables more efficient training and faster inference than existing models. Extensive experiments demonstrate that our model surpasses current operational NWP systems, extrapolation-based methods, and deep-learning nowcasting models, setting a new standard for high-resolution precipitation forecasting in Europe, ensuring a balance between accuracy, interpretability, and computational efficiency.

Applications · Physics

Alexander Rudikov, Vladimir Fanaskov, Sergei Stepanov, Buzheng Shan, Ekaterina Muravleva, Yalchin Efendiev, Ivan Oseledets

We propose GMsFEM-NO, a novel hybrid framework that combines the robustness of the Generalized Multiscale Finite Element Method (GMsFEM) with the computational speed of neural operators (NOs) to create an efficient method for solving heterogeneous partial differential equations (PDEs). GMsFEM builds localized spectral basis functions on coarse grids, allowing it to capture important multiscale features and solve PDEs accurately with less computational effort. However, computing these basis functions is costly. While NOs offer a fast alternative by learning the solution operator directly from data, they can lack robustness. Our approach trains a NO to instantly predict the GMsFEM basis by using a novel subspace-informed loss that learns the entire relevant subspace, not just individual functions. This strategy significantly accelerates the costly offline stage of GMsFEM while retaining its foundation in rigorous numerical analysis, resulting in a solution that is both fast and reliable. On standard multiscale benchmarks—including a linear elliptic diffusion problem and the nonlinear, steady-state Richards equation—our GMsFEM-NO method achieves a reduction in solution error compared to standalone NOs and other hybrid methods. The framework demonstrates effective performance for both 2D and 3D problems. A key advantage is its discretization flexibility: the NO can be trained on a small computational grid and evaluated on a larger one with minimal loss of accuracy, ensuring easy scalability. Furthermore, the resulting solver remains independent of forcing terms, preserving the generalization capabilities of the original GMsFEM approach. Our results prove that combining NO with GMsFEM creates a powerful new type of solver that is both fast and accurate.

Applications · Physics

Harris Abdul Majid, Pietro Sittoni, Francesco Tudisco

Accuracy-cost trade-offs are a fundamental aspect of scientific computing. Classical numerical methods inherently offer such a trade-off: increasing resolution, order, or precision typically yields more accurate solutions at higher computational cost. We introduce \textbf{Recurrent-Depth Simulator} (\textbf{RecurrSim}) an architecture-agnostic framework that enables explicit test-time control over accuracy-cost trade-offs in neural simulators without requiring retraining or architectural redesign. By setting the number of recurrent iterations $K$, users can generate fast, less-accurate simulations for exploratory runs or real-time control loops, or increase $K$ for more-accurate simulations in critical applications or offline studies. We demonstrate RecurrSim's effectiveness across fluid dynamics benchmarks (Burgers, Korteweg-De Vries, Kuramoto-Sivashinsky), achieving physically faithful simulations over long horizons even in low-compute settings. On high-dimensional 3D compressible Navier-Stokes simulations with 262k points, a 0.8B parameter RecurrFNO outperforms 1.6B parameter baselines while using 13.5\% less training memory. RecurrSim consistently delivers superior accuracy-cost trade-offs compared to alternative adaptive-compute models, including Deep Equilibrium and diffusion-based approaches. We further validate broad architectural compatibility: RecurrViT reduces error accumulation by 90\% compared to standard Vision Transformers on Active Matter, while RecurrUPT matches UPT performance on ShapeNet-Car using 44\% fewer parameters.

Reinforcement Learning · Deep RL

Sungbin Mun, Younghwan Lee, Cheolhui MIn, Mineui Hong, Young Min Kim

Vision-based reinforcement learning can benefit from object-centric scene representation, which factorizes the visual observation into individual objects and their attributes, such as color, shape, size, and position. While such object-centric representations can extract components that generalize well for various multi-object manipulation tasks, they are prone to issues with occlusions and 3D ambiguity of object properties due to their reliance on single-view 2D image features. Furthermore, the entanglement between object configurations and camera poses complicates the object-centric disentanglement in 3D, leading to poor 3D reasoning by the agent in vision-based reinforcement learning applications. To address the lack of 3D awareness and the object-camera entanglement problem, we propose an enhanced 3D object-centric representation that utilizes multi-view 3D features and enforces more explicit 3D-aware disentanglement. The enhancement is based on the integration of the recent success of multi-view Transformer and the prototypical representation learning among the object-centric representations. The representation, therefore, can stably identify proxies of 3D positions of individual objects along with their semantic and physical properties, exhibiting excellent interpretability and controllability. Then, our proposed block transformer policy effectively performs novel tasks by assembling desired properties adaptive to the new goal states, even when provided with unseen viewpoints at test time. We demonstrate that our 3D-aware block representation is scalable to compose diverse novel scenes and enjoys superior performance in out-of-distribution tasks with multi-object manipulations under both seen and unseen viewpoints compared to existing methods.

Applications · Physics

Yuxin Li, Minghao LIU, Ruida WANG, JI WenZhao, Zhitao He, Rui Pan, Junming Huang, Tong Zhang, Yi R. Fung

We present **Lean4PHYS**, a comprehensive reasoning framework for college-level physics problems in Lean4. To establish a solid foundation for formal reasoning in physics, **Lean4PHYS** launches *PhysLib*, a repository containing fundamental unit systems and essential theorems to formulate physics proofs in Lean4. It will be community-driven and long-term maintained. Lean4PHYS also includes *LeanPhysBench*, a college-level benchmark for evaluating LLMs' Lean4 formal physics reasoning capability. It contains 200 hand-crafted and peer-reviewed Lean4 theorem statements formalized from university textbooks and physics competition problems. Based on the *PhysLib* and *LeanPhysBench* we composed in **Lean4PHYS**, we perform exhaustive experiments of baseline results using major expert Math provers and state-of-the-art closed-source models, and provide an analysis of their performance. In the experiment, we identify that most expert provers do not outperform general models as they did in the math domain. This suggests potential overfitting to the math domain rather than learning formal reasoning for formal provers. We also conduct a comprehensive experiment showing that, with *PhysLib* in the context, LLMs' performance on *LeanPhysBench* increases by **11.90%** on average, proving the effectiveness of our repository in assisting LLMs in solving the Lean4 physics problem. To the best of our knowledge, we are the first study to provide a physics benchmark in Lean4.

Applications · Robotics

Xupeng Zhu, Yu Qi, Yizhe Zhu, Robin Walters, Robert Platt

Multi-task manipulation policy often builds on transformer's ability to jointly process language instructions and 3D observations in a shared embedding space. However, real-world tasks frequently require robots to generalize to novel 3D object poses. Policies based on shared embedding break geometric consistency and struggle in 3D generation. To address this issue, we propose EquAct, which is theoretically guaranteed to generalize to novel 3D scene transformations by leveraging SE(3) equivariance shared across both language, observations, and action. EquAct makes two key contributions: (1) an efficient SE(3)-equivariant point cloud-based U-net with spherical Fourier features for policy reasoning, and (2) SE(3)-invariant Feature-wise Linear Modulation (iFiLM) layers for language conditioning. Finally, EquAct demonstrates strong spatial generalization ability and achieves state-of-the-art across $18$ RLBench tasks with both SE(3) and SE(2) scene perturbations, different amounts of training data, and on $4$ physical tasks.

Applications · Chemistry and Drug Discovery

Danyal Rehman, Tara Akhound-Sadegh, Artem Gazizov, Yoshua Bengio, Alexander Tong

Scalable sampling of molecular states in thermodynamic equilibrium is a long-standing challenge in statistical physics. Boltzmann Generators tackle this problem by pairing a generative model, capable of exact likelihood computation, with importance sampling to obtain consistent samples under the target distribution. Current Boltzmann Generators primarily use continuous normalizing flows (CNFs) trained with flow matching for efficient training of powerful models. However, likelihood calculation for these models is extremely costly, requiring thousands of function evaluations per sample, severely limiting their adoption. In this work, we propose Few-Step Accurate Likelihoods for Continuous Flows (FALCON), a method which allows for few-step sampling with a likelihood accurate enough for importance sampling applications by introducing a hybrid training objective that encourages invertibility. We show FALCON outperforms state-of-the-art normalizing flow models for molecular Boltzmann sampling and is two orders of magnitude faster than the equivalently performing CNF model. FALCON code is available at: https://github.com/danyalrehman/FALCON.

Optimization · Sampling and Optimization

Wei Guo, Jaemoo Choi, Yuchen Zhu, Molei Tao, Yongxin Chen

The task of learning a diffusion-based neural sampler for drawing samples from an unnormalized target distribution can be viewed as a stochastic optimal control problem on path measures. However, the training of neural samplers can be challenging when the target distribution is multimodal with significant barriers separating the modes, potentially leading to mode collapse. We propose a framework named **Proximal Diffusion Neural Sampler (PDNS)** that addresses these challenges by tackling the stochastic optimal control problem via proximal point method on the space of path measures. PDNS decomposes the learning process into a series of simpler subproblems that create a path gradually approaching the desired distribution. This staged procedure traces a progressively refined path to the desired distribution and promotes thorough exploration across modes. For a practical and efficient realization, we instantiate each proximal step with a proximal weighted denoising cross-entropy (WDCE) objective. We demonstrate the effectiveness and robustness of PDNS through extensive experiments on both continuous and discrete sampling tasks, including challenging scenarios in molecular dynamics and statistical physics. Our code is available at https://github.com/AlexandreGUO2001/PDNS.

Applications · Physics

Utkarsh Utkarsh, Danielle Maddix, Ruijun Ma, Michael W Mahoney, Bernie Wang

We present ProbHardE2E, a probabilistic forecasting framework that incorporates hard operational/physical constraints and provides uncertainty quantification. Our methodology uses a novel differentiable probabilistic projection layer (DPPL) that can be combined with a wide range of neural network architectures. DPPL allows the model to learn the system in an end-to-end manner, compared to other approaches where constraints are satisfied either through a post-processing step or at inference. ProbHardE2E optimizes a strictly proper scoring rule, without making any distributional assumptions on the target, which enables it to obtain robust distributional estimates (in contrast to existing approaches that generally optimize likelihood-based objectives, which can be biased by their distributional assumptions and model choices); and it can incorporate a range of non-linear constraints (increasing the power of modeling and flexibility). We apply ProbHardE2E in learning partial differential equations with uncertainty estimates and to probabilistic time-series forecasting, showcasing it as a broadly applicable general framework that connects these seemingly disparate domains. Our code is available at https://github.com/amazon-science/probharde2e.

Applications · Chemistry and Drug Discovery

Nima Shoghi, Yuxuan Liu, Yuning Shen, Rob Brekelmans, Pan Li, Quanquan Gu

Molecular dynamics (MD) simulations remain the gold standard for studying protein dynamics, but their computational cost limits access to biologically relevant timescales. Recent generative models have shown promise in accelerating simulations, yet they struggle with long-horizon generation due to architectural constraints, error accumulation, and inadequate modeling of spatiotemporal dynamics. We present STAR-MD (Spatio-Temporal Autoregressive Rollout for Molecular Dynamics), a scalable SE(3)-equivariant diffusion model that generates physically plausible protein trajectories over microsecond timescales. Our key innovation is a causal diffusion transformer with joint spatiotemporal attention that efficiently captures complex space-time dependencies while avoiding the memory bottlenecks of existing methods. On the standard ATLAS benchmark, STAR-MD achieves state-of-the-art performance across all metrics--substantially improving conformational coverage, structural validity, and dynamic fidelity compared to previous methods. STAR-MD successfully extrapolates to generate stable microsecond-scale trajectories where baseline methods fail catastrophically, maintaining high structural quality throughout the extended rollout. Our comprehensive evaluation reveals severe limitations in current models for long-horizon generation, while demonstrating that STAR-MD's joint spatiotemporal modeling enables robust dynamics simulation at biologically relevant timescales, paving the way for accelerated exploration of protein function. Project page: https://bytedance-seed.github.io/ConfRover/starmd