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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

Applications · Chemistry and Drug Discovery

Richard Strunk, Karnik Ram, Daniel Cremers

MLIPs are a promising paradigm in atomistic simulation, potentially offering the accuracy of ab-initio methods at the speed of empirical potentials. In this blog post, we give an overview of recent MLIP architectures, followed by an evaluation on a practical CO2 adsorption simulation. We find that as of today these models, though promising, are far from plug-and-play, requiring significant engineering effort to operate within established simulation frameworks, while also failing to produce physically consistent results.

Deep Learning · Graph Neural Networks

Maksim Zhdanov, Nabil Iqbal, Erik Bekkers, Patrick Forré

Conformal symmetries, i.e.\ coordinate transformations that preserve angles, play a key role in many fields, including physics, mathematics, computer vision and (geometric) machine learning. Here we build a neural network that is equivariant under general conformal transformations. To achieve this, we lift data from flat Euclidean space to Anti de Sitter (AdS) space. This allows us to exploit a known correspondence between conformal transformations of flat space and isometric transformations on the Anti de Sitter space. We then build upon the fact that such isometric transformations have been extensively studied on general geometries in the geometric deep learning literature. In particular, we employ message-passing layers conditioned on the proper distance, yielding a computationally efficient framework. We validate our model on tasks from computer vision and statistical physics, demonstrating strong performance, improved generalization capacities, and the ability to extract conformal data such as scaling dimensions from the trained network.

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

Seokki Lee, Min-Chul Park, Giyong Hong, Changwook Jeong

PDE surrogate models such as FNO and PINN struggle to predict solutions across inputs with diverse physical units and scales, limiting their out-of-distribution (OOD) generalization. We propose a $\pi$-invariant test-time projection that aligns test inputs with the training distribution by solving a log-space least squares problem that preserves Buckingham $\pi$-invariants. For PDEs with multidimensional spatial fields, we use geometric representative $\pi$-values to compute distances and project inputs, overcoming degeneracy and singular points that limit prior $\pi$-methods. To accelerate projection, we cluster the training set into K clusters, reducing the complexity from O(MN) to O(KN) for the M training and N test samples. Across wide input scale ranges, tests on 2D thermal conduction and linear elasticity achieve an average MAE reduction up to $\approx 91\\%$ with minimal overhead. This training-free, model-agnostic method is expected to apply to more diverse PDE-based simulations.

Reinforcement Learning · Batch/Offline

Perry Dong, Chongyi Zheng, Chelsea Finn, Dorsa Sadigh, Benjamin Eysenbach

While most reinforcement learning methods today flatten the distribution of future returns to a single scalar value, distributional RL methods exploit the return distribution to provide stronger learning signals and to enable applications in exploration and safe RL. While the predominant method for estimating the return distribution is by modeling it as a categorical distribution over discrete bins or estimating a finite number of quantiles, such approaches leave unanswered questions about the fine-grained structure of the return distribution and about how to distinguish states with high return uncertainty for decision-making. The key idea in this paper is to use modern, flexible flow-based models to estimate the full future return distributions and identify those states with high return variance. We do so by formulating a new flow-matching objective that generates probability density paths satisfying the distributional Bellman equation. Building upon the learned flow models, we estimate the return uncertainty of distinct states using a new flow derivative ODE. We additionally use this uncertainty information to prioritize learning a more accurate return estimation on certain transitions. We compare our method (Value Flows) with prior methods in the offline and online-to-online settings. Experiments on $37$ state-based and $25$ image-based benchmark tasks demonstrate that Value Flows achieves a $1.3\times$ improvement on average in success rates.

Computer Vision · 3D Rendering & Reconstruction

Yang Zhou, Yifan Wang, Jianjun Zhou, Wenzheng Chang, Haoyu Guo, Zizun Li, Kaijing Ma, Xinyue Li, Yating Wang, Haoyi Zhu 等

The field of 4D world modeling—aiming to jointly capture spatial geometry and temporal dynamics—has witnessed remarkable progress in recent years, driven by advances in large-scale generative models and multimodal learning. However, the development of truly general 4D world models remains fundamentally constrained by the availability of high-quality data. Existing datasets and benchmarks often lack the dynamic complexity, multi-domain diversity, and spatial-temporal annotations required to support key tasks such as 4D geometric reconstruction, future prediction, and camera-controlled video generation. To address this gap, we introduce OmniWorld, a large-scale, multi-domain, multi-modal dataset specifically designed for 4D world modeling. OmniWorld consists of a newly collected OmniWorld-Game dataset and several curated public datasets spanning diverse domains. Compared with existing synthetic datasets, OmniWorld-Game provides richer modality coverage, larger scale, and more realistic dynamic interactions. Based on this dataset, we establish a challenging benchmark that exposes the limitations of current state-of-the-art (SOTA) approaches in modeling complex 4D environments. Moreover, fine-tuning existing SOTA methods on OmniWorld leads to significant performance gains across 4D reconstruction and video generation tasks, strongly validating OmniWorld as a powerful resource for training and evaluation. We envision OmniWorld as a catalyst for accelerating the development of general-purpose 4D world models, ultimately advancing machines’ holistic understanding of the physical world.

Applications · Neuroscience, Cognitive Science

Balázs Meszéna, Keith Murray, Julien Corbo, Batuhan Erkat, Márton Hajnal, Pierre-Olivier Polack, Gergo Orban

The brain interprets visual information through learned regularities, a computation formalized as performing probabilistic inference under a prior. The visual cortex establishes priors for this inference, some of which are delivered through widely established top-down connections that inform low-level cortices about statistics represented at higher levels in the cortical hierarchy. While evidence supports that adaptation leads to priors reflecting the structure of natural images, it remains unclear if similar priors can be flexibly acquired when learning a specific task. To investigate this, we built a generative model of V1 that we optimized for performing a simple discrimination task and analyzed it along with large scale recordings from mice performing an analogous task. In line with recent successful approaches, we assumed that neuronal activity in V1 can be identified with latent posteriors in the generative model, providing an opportunity to investigate the contributions of task-related priors to neuronal responses. To obtain a flexible test bed for this analysis, we extended the VAE formalism so that a task can be flexibly and data-efficiently acquired by reusing previously learned representations. Task-specific priors learned by this Task-Amortized VAE were used to investigate biases in mice and model when presenting stimuli that violated the trained task statistics. Mismatch between learned task statistics and incoming sensory evidence showed signatures of uncertainty in stimulus category in the posterior of TAVAE, reflecting properties of bimodal response profile in V1 recordings. The task-optimized generative model could account for various characteristics of V1 population activity, including within-day updates to the population responses. Our results confirm that flexible task-specific contextual priors can be learned on-demand by the visual system and can be deployed as early as the entry level of the visual cortex.

Deep Learning · Generative Models and Autoencoders

Janik Kreit, Dominic Schuh, Kim A. Nicoli, Lena Funcke

Deep generative models have recently garnered significant attention across various fields, from physics to chemistry, where sampling from unnormalized Boltzmann-like distributions represents a fundamental challenge. In particular, autoregressive models and normalizing flows have become prominent due to their appealing ability to yield closed-form probability densities. Moreover, it is well-established that incorporating prior knowledge—such as symmetries—into deep neural networks can substantially improve training performances. In this context, recent advances have focused on developing symmetry-equivariant generative models, achieving remarkable results. Building upon these foundations, this paper introduces Symmetry-Enforcing Stochastic Modulation (SESaMo). Similar to equivariant normalizing flows, SESaMo enables the incorporation of inductive biases (e.g., symmetries) into normalizing flows through a novel technique called \textit{stochastic modulation}. This approach enhances the flexibility of the generative model by enforcing exact symmetries while, for the first time, enabling the model to learn broken symmetries during training. Our numerical experiments benchmark SESaMo in different scenarios, including an 8-Gaussian mixture model and physically relevant field theories, such as the $\phi^4$ theory and the Hubbard model.

Applications · Physics

Jan Tauberschmidt, Sophie Fellenz, Sebastian Vollmer, Andrew Duncan

We present a framework for fine-tuning flow-matching generative models to enforce physical constraints and solve inverse problems in scientific systems. Starting from a model trained on low-fidelity or observational data, we apply a differentiable post-training procedure that minimizes weak-form residuals of governing partial differential equations (PDEs), promoting physical consistency and adherence to boundary conditions without distorting the underlying learned distribution. To infer unknown physical inputs, such as source terms, material parameters, or boundary data, we augment the generative process with a learnable latent parameter predictor and propose a joint optimization strategy. The resulting model produces physically valid field solutions alongside plausible estimates of hidden parameters, effectively addressing ill-posed inverse problems in a data-driven yet physics-aware manner. We validate our method on canonical PDE problems, demonstrating improved satisfaction of physical constraints and accurate recovery of latent coefficients. Further, we confirm cross-domain utility through fine-tuning of natural-image models. Our approach bridges generative modelling and scientific inference, opening new avenues for simulation-augmented discovery and data-efficient modelling of physical systems.

Deep Learning · Generative Models and Autoencoders

Xinpeng Li, Enming Liang, Minghua Chen

Generative models, particularly diffusion and flow-matching approaches, have achieved remarkable success across diverse domains, including image synthesis and robotic planning. However, a fundamental challenge persists: ensuring generated samples strictly satisfy problem-specific constraints — a crucial requirement for physics-informed problems, safety-critical applications, watermark embedding, etc. Existing approaches, such as mirror maps and reflection methods, either have limited applicable constraint sets or introduce significant computational overhead. In this paper, we develop gauge flow matching (GFM), a simple yet efficient framework for constrained generative modeling. Our GFM approach introduces a novel bijective gauge mapping to transform generation over arbitrary compact convex sets into an equivalent process over the unit ball, which allows low-complexity feasibility-ensuring operations such as reflection or projection. The generated samples are then mapped back to the original domain for output. We prove that our GFM framework guarantees strict constraint satisfaction, with low generation complexity and bounded distribution approximation errors. We further extend our GFM framework to two non-convex settings, namely, star-convex and geodesic-convex sets. Extensive experiments demonstrate that GFM outperforms existing methods in both generation speed and quality across multiple benchmarks, including synthetic data, time series, and image generation.

Deep Learning · Theory

Yuxi Liu, Konpat Preechakul, Kananart Kuwaranancharoen, Yutong Bai

While machine learning has advanced through massive parallelization, we identify a critical blind spot: some problems are fundamentally sequential. These "inherently serial" problems—from mathematical reasoning to physical simulations to sequential decision-making—require sequentially dependent computational steps that cannot be efficiently parallelized. We formalize this distinction in complexity theory, and demonstrate that current parallel-centric architectures face fundamental limitations on such tasks. Then, we show for first time that diffusion models despite their sequential nature are incapable of solving inherently serial problems. We argue that recognizing the serial nature of computation holds profound implications on machine learning, model design, and hardware development.

Applications · Physics

Zihan Zhou, Chenguang Wang, Hongyi Ye, Yongtao Guan, Tianshu Yu

Learning physical dynamics from data is a fundamental challenge in machine learning and scientific modeling. Real-world observational data are inherently incomplete and irregularly sampled, posing significant challenges for existing data-driven approaches. In this work, we propose a principled diffusion-based framework for learning physical systems from incomplete training samples. To this end, our method strategically partitions each such sample into observed context and unobserved query components through a carefully designed splitting strategy, then trains a conditional diffusion model to reconstruct the missing query portions given available contexts. This formulation enables accurate imputation across arbitrary observation patterns without requiring complete data supervision. Specifically, we provide theoretical analysis demonstrating that our diffusion training paradigm on incomplete data achieves asymptotic convergence to the true complete generative process under mild regularity conditions. Empirically, we show that our method significantly outperforms existing baselines on synthetic and real-world physical dynamics benchmarks, including fluid flows and weather systems, with particularly strong performance in limited and irregular observation regimes. These results demonstrate the effectiveness of our theoretically principled approach for learning and imputing partially observed dynamics.

Deep Learning · Everything Else

Jaeyoung Hong, YunYoung Choi, Joohwan Ko, Minseon Gwak

Modeling long spatiotemporal sequences requires capturing both complex spatial correlations and temporal dependencies. Convolutional State Space Models (ConvSSMs) have been proposed to incorporate spatial modeling in State Space Models (SSMs) using the convolution of tensor-valued states and kernels. Yet, existing implementations remain limited to $1\times 1$ state kernels for computational feasibility, which limits the modeling capacity of ConvSSMs. We introduce a novel spatiotemporal model, ConvT3 (ConvSSM using Tridiagonal Toeplitz Tensors), designed to equivalently realize ConvSSMs with extended $3\times 3$ state kernels. ConvT3 structures a state kernel for its corresponding tensor to be composed as a structured SSM matrix on hidden state dimensions and a constrained tridiagonal Toeplitz tensor on spatial dimensions. We show that the structured tensor can be diagonalized, which enables efficient parallel training while leveraging $3\times 3$ state convolutions. We demonstrate that ConvT3 effectively embeds rich spatial and temporal information into the dynamics of tensor-valued states, achieving state-of-the-art performance on most metrics in long-range video generation and physical system modeling.

Computer Vision · Image and Video Generation

Jianhao Yuan, Fabio Pizzati, Francesco Pinto, Lars Kunze, Ivan Laptev, Paul Newman, Philip Torr, Daniele De Martini

Intuitive physics understanding in video diffusion models plays an essential role in building general-purpose physically plausible world simulators, yet accurately evaluating such capacity remains a challenging task due to the difficulty in disentangling physics correctness from visual appearance in generation. To the end, we introduce LikePhys, a training-free method that evaluates intuitive physics in video diffusion models by distinguishing physically valid and impossible videos using the denoising objective as an ELBO-based likelihood surrogate on a curated dataset of valid-invalid pairs. By testing on our constructed benchmark of twelve scenarios spanning over four physics domains, we show that our evaluation metric, Plausibility Preference Error (PPE), demonstrates strong alignment with human preference, outperforming state-of-the-art evaluator baselines. We then systematically benchmark intuitive physics understanding in current video diffusion models. Our study further analyses how model design and inference settings affect intuitive physics understanding and highlights domain-specific capacity variations across physical laws. Empirical results show that, despite current models struggling with complex and chaotic dynamics, there is a clear trend of improvement in physics understanding as model capacity and inference settings scale up.

Computer Vision · Vision Models & Multimodal

Anbu Huang

This blogpost develops a geometric and probabilistic lens on diffusion priors for inverse problems. We show that a wide range of methods mostly instantiate two operator-splitting paradigms, i.e., posterior-guided sampling and clean-space local-MAP optimization. Through manifold diagrams, Tweedie-based animations, and step-by-step derivations, we explain how these paradigms decouple a pretrained diffusion prior from measurement physics, clarify when they approximate full posterior sampling versus MAP estimation, and distill practical design rules for building robust diffusion-based inverse solvers.

Applications · Physics

Yingjie Tan, Quanming Yao, Yaqing Wang

Spatiotemporal partial differential equations (PDEs) underpin a wide range of scientific and engineering applications. Neural PDE solvers offer a promising alternative to classical numerical methods. However, existing approaches typically require large numbers of training trajectories, while high-fidelity PDE data are expensive to generate. Under limited data, their performance degrades substantially, highlighting their low data efficiency. A key reason is that PDE dynamics embody strong structural inductive biases that are not explicitly encoded in neural architectures, forcing models to learn fundamental physical structure from data. A particularly salient manifestation of this inefficiency is poor generalization to unseen source terms. In this work, we revisit Green’s function theory—a cornerstone of PDE theory—as a principled source of structural inductive bias for PDE learning. Based on this insight, we propose DGNet, a discrete Green network for data-efficient learning of spatiotemporal PDEs. The key idea is to transform the Green’s function into a graph-based discrete formulation, and embed the superposition principle into the hybrid physics–neural architecture which reduces the burden of learning physical priors from data, thereby improving sample efficiency. Across diverse spatiotemporal PDE scenarios, DGNet consistently achieves state-of-the-art accuracy using only tens of training trajectories. Moreover, it exhibits robust zero-shot generalization to unseen source terms, serving as a stress test that highlights its data-efficient structural design.

Applications · Physics

Haining Pan, James Roggeveen, Erez Berg, Juan Alvarez, Debanjan Chowdhury, Surya Ganguli, Federico Ghimenti, Juraj Hasik, Henry Hunt, Hong-Chen Jiang 等

Large language models (LLMs) have demonstrated remarkable progress in coding and mathematical problem-solving; however, evaluation on advanced research-level problems in the hard sciences remains scarce. To fill this gap, we present \cmt, a dataset of 50 original problems covering condensed matter theory (CMT) at the level of an expert researcher. The solution for these problems involve analytical and computational approaches commonly used in quantum many-body physics and classical statistical mechanics. The dataset has been designed and verified by a worldwide panel of expert researchers through a collaborative environment. Topics in the dataset include Hartree-Fock mean-field theory, exact diagonalization methods, quantum Monte Carlo sampling, density matrix renormalization group, quantum statistical mechanics, classical statistical mechanics, and model building. We evaluate different LLMs by programmatically checking LLM-generated solutions against expert-supplied ground truth. To verify LLMs performance at scale, we developed an automated machine-grading pipeline suitable for advanced physics research problems. For example, we handle non-commuting operators that are essential for quantum many-body problems by symbolic manipulation and normal ordering. Our evaluations show that frontier models struggle with all of the problems in the dataset, highlighting a gap in the physical reasoning skills of current LLMs. Notably, experts identified strategies for creating increasingly difficult problems by interacting with the LLMs and exploiting common failure modes. While the highest-performing model, GPT5, correctly solves 30\% of the problems, average performance across 17 models (GPT, Gemini, Claude, DeepSeek, and Llama classes) is only 11.4$\pm$2.1\%. Moreover, our benchmark contains 18 problems that not a single one of the 17 models considered here can correctly solve, and 26 problems that are solved by at most one model. These currently unsolvable problems span the fields of Quantum Monte Carlo, Variational Monte Carlo, and Density Matrix Renormalization Group. Furthermore, we illustrate how incorrect answers sometimes violate fundamental symmetries or have unphysical scaling dimensions. We believe that this benchmark set provides valuable guidance for the future development of language models, aiming to achieve the goal of AI research assistants and tutors.

Applications · Physics

Tai Hoang, Alessandro Trenta, Alessio Gravina, Niklas Freymuth, Philipp Becker, Davide Bacciu, Gerhard Neumann

Learning to simulate complex physical systems from data has emerged as a promising way to overcome the limitations of traditional numerical solvers, which often require prohibitive computational costs for high-fidelity solutions. Recent Graph Neural Simulators (GNSs) accelerate simulations by learning dynamics on graph-structured data, yet often struggle to capture long-range interactions and suffer from error accumulation under autoregressive rollouts. To address these challenges, we propose Information-preserving Graph Neural Simulators (IGNS), a graph-based neural simulator built on the principles of Hamiltonian dynamics. This structure guarantees preservation of information across the graph, while extending to port-Hamiltonian systems allows the model to capture a broader class of dynamics, including non-conservative effects. IGNS further incorporates a warmup phase to initialize global context, geometric encoding to handle irregular meshes, and a multi-step training objective that facilitates PDE matching, where the trajectory produced by integrating the port-Hamiltonian core aligns with the ground-truth trajectory, thereby reducing rollout error. To evaluate these properties systematically, we introduce new benchmarks that target long-range dependencies and challenging external forcing scenarios. Across all tasks, IGNS consistently outperforms state-of-the-art GNSs, achieving higher accuracy and stability under challenging and complex dynamical systems. Our project page: https://thobotics.github.io/neural_pde_matching.