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Applications · Robotics

Yunke Ao, Masoud Moghani, Mayank Mittal, Manish Prajapat, Luohong Wu, Frederic Giraud, Fabio Carrillo, Andreas Krause, Philipp Fürnstahl

Ultrasound (US) is a widely used medical imaging modality due to its real-time capabilities, non-invasive nature, and cost-effectiveness. By reducing operator dependency and enhancing access to complex anatomical regions, robotic ultrasound can help improve workflow efficiency. Recent studies have demonstrated the potential of deep reinforcement learning (DRL) and imitation learning (IL) to enable more autonomous and intelligent robotic ultrasound navigation. However, the application of learning-based robotic ultrasound to computer-assisted surgical tasks, such as anatomy reconstruction and surgical guidance, remains largely unexplored. A key bottleneck for this is the lack of realistic and efficient simulation environments tailored to these tasks. In this work, we present SonoGym, a scalable simulation platform for robotic ultrasound, enabling parallel simulation across tens to hundreds of environments. Our framework supports realistic and real-time simulation of US data from CT-derived 3D models of the anatomy through both a physics-based and a Generative Adversarial Network (GAN) approach. Our framework enables the training of DRL and recent IL agents (vision transformers and diffusion policies) for relevant tasks in robotic orthopedic surgery by integrating common robotic platforms and orthopedic end effectors. We further incorporate submodular DRL---a recent method that handles history-dependent rewards---for anatomy reconstruction and safe reinforcement learning for surgery. Our results demonstrate successful policy learning across a range of scenarios, while also highlighting the limitations of current methods in clinically relevant environments. We believe our simulation can facilitate research in robot learning approaches for such challenging robotic surgery applications. Dataset, codes and videos are publicly available at https://sonogym.github.io/.

Applications · Physics

Shiqian Li, Zhi Li, Zhancun Mu, Shiji Xin, Zhixiang Dai, Kuangdai Leng, Rita Zhang, Xiaodong Song, Yixin Zhu

Global seismic tomography, taking advantage of seismic waves from natural earthquakes, provides essential insights into the earth's internal dynamics. Advanced Full-Waveform Inversion (FWI) techniques, whose aim is to meticulously interpret every detail in seismograms, confront formidable computational demands in forward modeling and adjoint simulations on a global scale. Recent advancements in Machine Learning (ML) offer a transformative potential for accelerating the computational efficiency of FWI and extending its applicability to larger scales. This work presents the first 3D global synthetic dataset tailored for seismic wavefield modeling and full-waveform tomography, referred to as the Global Tomography (GlobalTomo) dataset. This dataset is comprehensive, incorporating explicit wave physics and robust geophysical parameterization at realistic global scales, generated through state-of-the-art forward simulations optimized for 3D global wavefield calculations. Through extensive analysis and the establishment of ML baselines, we illustrate that ML approaches are particularly suitable for global FWI, overcoming its limitations with rapid forward modeling and flexible inversion strategies. This work represents a cross-disciplinary effort to enhance our understanding of the earth's interior through physics-ML modeling.

Applications

Yaoyao Xu, Di Wang, Zihan Zhou, Tianshu Yu, Mingchen Chen

Understanding the dynamic behavior of proteins is critical to elucidating their functional mechanisms, yet generating realistic, temporally coherent trajectories of protein ensembles remains a significant challenge. In this work, we introduce a novel hierarchical autoregressive framework for modeling protein dynamics that leverages the intrinsic multi-scale organization of molecular motions. Unlike existing methods that focus on generating static conformational ensembles or treat dynamic sampling as an independent process, our approach characterizes protein dynamics as a Markovian process. The framework employs a two-scale architecture: a low-resolution model captures slow, collective motions driving major conformational transitions, while a high-resolution model generates detailed local fluctuations conditioned on these large-scale movements. This hierarchical design ensures that the causal dependencies inherent in protein dynamics are preserved, enabling the generation of temporally coherent and physically realistic trajectories. By bridging high-level biophysical principles with state-of-the-art generative modeling, our approach provides an efficient framework for simulating protein dynamics that balances computational efficiency with physical accuracy.

Optimization · Neural Network Optimization

Yuhao Zhou, Jintao Xu, Bingrui Li, Chenglong Bao, Chao Ding, Jun Zhu

Finding an $\epsilon$-stationary point of a nonconvex function with a Lipschitz continuous Hessian is a central problem in optimization. Regularized Newton methods are a classical tool and have been studied extensively, yet they still face a trade‑off between global and local convergence. Whether a parameter-free algorithm of this type can simultaneously achieve optimal global complexity and quadratic local convergence remains an open question. To bridge this long-standing gap, we propose a new class of regularizers constructed from the current and previous gradients, and leverage the conjugate gradient approach with a negative curvature monitor to solve the regularized Newton equation. The proposed algorithm is adaptive, requiring no prior knowledge of the Hessian Lipschitz constant, and achieves a global complexity of $O(\epsilon^{-\frac{3}{2}})$ in terms of the second-order oracle calls, and $\tilde O(\epsilon^{-\frac{7}{4}})$ for Hessian-vector products, respectively. When the iterates converge to a point where the Hessian is positive definite, the method exhibits quadratic local convergence. Preliminary numerical results, including training the physics-informed neural networks, illustrate the competitiveness of our algorithm.

Fan LIU, Zherui Yang, Cancheng Liu, Tianrui Song, Xiaofeng Gao, Hao Liu

Mathematical modeling is a cornerstone of scientific discovery and engineering practice, enabling the translation of real-world problems into formal systems across domains such as physics, biology, and economics. Unlike mathematical reasoning, which assumes a predefined formulation, modeling requires open-ended problem analysis, abstraction, and principled formalization. While Large Language Models (LLMs) have shown strong reasoning capabilities, they fall short in rigorous model construction, limiting their utility in real-world problem-solving. To this end, we formalize the task of LLM-powered real-world mathematical modeling, where agents must analyze problems, construct domain-appropriate formulations, and generate complete end-to-end solutions.We introduce MM-Bench, a curated benchmark of 111 problems from the Mathematical Contest in Modeling (MCM/ICM), spanning the years 2000 to 2025 and across ten diverse domains such as physics, biology, and economics. To tackle this task, we propose MM-Agent, an expert-inspired framework that decomposes mathematical modeling into four stages: open-ended problem analysis, structured model formulation, computational problem solving, and report generation.Experiments on MM-Bench show that MM-Agent significantly outperforms baseline agents, achieving an 11.88\% improvement over human expert solutions while requiring only 15 minutes and \$0.88 per task using GPT-4o. Furthermore, under official MCM/ICM protocols, MM-Agent assisted two undergraduate teams in winning the Finalist Award (\textbf{top 2.0\% among 27,456 teams}) in MCM/ICM 2025, demonstrating its practical effectiveness as a modeling copilot.

Applications · Neuroscience, Cognitive Science

Qinfan Xiao, Ziyun Cui, Chi Zhang, SiQi Chen, Wen Wu, Andrew Thwaites, Alexandra Woolgar, Bowen Zhou, Chao Zhang

Electroencephalography (EEG) and magnetoencephalography (MEG) measure neural activity non-invasively by capturing electromagnetic fields generated by dendritic currents. Although rooted in the same biophysics, EEG and MEG exhibit distinct signal patterns, further complicated by variations in sensor configurations across modalities and recording devices. Existing approaches typically rely on separate, modality- and dataset-specific models, which limits the performance and cross-domain scalability. This paper proposes BrainOmni, the first brain foundation model that generalises across heterogeneous EEG and MEG recordings. To unify diverse data sources, we introduce BrainTokenizer, the first tokeniser that quantises spatiotemporal brain activity into discrete representations. Central to BrainTokenizer is a novel Sensor Encoder that encodes sensor properties such as spatial layout, orientation, and type, enabling compatibility across devices and modalities. Building upon the discrete representations, BrainOmni learns unified semantic embeddings of brain signals by self-supervised pretraining. To the best of our knowledge, it is the first foundation model to support both EEG and MEG signals, as well as the first to incorporate large-scale MEG pretraining. A total of 1,997 hours of EEG and 656 hours of MEG data are curated and standardised from publicly available sources for pretraining. Experiments show that BrainOmni outperforms both existing foundation models and state-of-the-art task-specific models on a range of downstream tasks. It also demonstrates strong generalisation to unseen EEG and MEG devices. Further analysis reveals that joint EEG-MEG (EMEG) training yields consistent improvements across both modalities. Code and checkpoints are publicly available at https://github.com/OpenTSLab/BrainOmni

Applications · Robotics

Yifan Xie, Mingyang Li, Shoujie Li, Xingting Li, Guangyu Chen, Fei Ma, Fei Yu, Wenbo Ding

Tactile perception is essential for embodied agents to understand the physical attributes of objects that cannot be determined through visual inspection alone. While existing methods have made progress in visual and language modalities for physical understanding, they fail to effectively incorporate tactile information that provides crucial haptic feedback for real-world interaction. In this paper, we present VTV-LLM, the first multi-modal large language model that enables universal Visuo-Tactile Video (VTV) understanding, bridging the gap between tactile perception and natural language. To address the challenges of cross-sensor and cross-modal integration, we contribute VTV150K, a comprehensive dataset comprising 150,000 video frames from 100 diverse objects captured across three different tactile sensors (GelSight Mini, DIGIT, and Tac3D), annotated with four fundamental tactile attributes (hardness, protrusion, elasticity, and friction). We develop a novel three-stage training paradigm that includes VTV enhancement for robust visuo-tactile representation, VTV-text alignment for cross-modal correspondence, and text prompt finetuning for natural language generation. Our framework enables sophisticated tactile reasoning capabilities including feature assessment, comparative analysis, and scenario-based decision-making. Extensive experimental evaluations demonstrate that VTV-LLM achieves superior performance in tactile reasoning tasks, establishing a foundation for more intuitive human-machine interaction in tactile domains.

Applications · Physics

Akash Kundu, Stefano Mangini

Variational quantum algorithms hold the promise to address meaningful quantum problems already on noisy intermediate-scale quantum hardware. In spite of the promise, they face the challenge of designing quantum circuits that both solve the target problem and comply with device limitations. Quantum architecture search (QAS) automates the design process of quantum circuits, with reinforcement learning (RL) emerging as a promising approach. Yet, RL-based QAS methods encounter significant scalability issues, as computational and training costs grow rapidly with the number of qubits, circuit depth, and hardware noise. To address these challenges, we introduce $\textit{TensorRL-QAS}$, {an improved} framework that combines tensor network methods with RL for QAS. By warm-starting the QAS with a matrix product state approximation of the target solution, TensorRL-QAS effectively narrows the search space to physically meaningful circuits and accelerates the convergence to the desired solution. Tested on several quantum chemistry problems of up to 12-qubit, TensorRL-QAS achieves up to a 10-fold reduction in CNOT count and circuit depth compared to baseline methods, while maintaining or surpassing chemical accuracy. It reduces classical optimizer function evaluation by up to 100-fold, accelerates training episodes by up to 98\%, and can achieve 50\% success probability for 10-qubit systems, far exceeding the $<$1\% rates of baseline. Robustness and versatility are demonstrated both in the noiseless and noisy scenarios, where we report a simulation of an 8-qubit system. Furthermore, TensorRL-QAS demonstrates effectiveness on systems on 20-qubit quantum systems, positioning it as a state-of-the-art quantum circuit discovery framework for near-term hardware and beyond.

Applications · Physics

Hongquan He, Zhen Wang, Jingya Wang, Tao Wu, Xuming He, Bei Yu, Jingyi Yu, Hao GENG

Lithography orchestrates a symphony of light, mask and photochemicals to transfer the integrated circuit patterns onto the wafer. Lithography simulation serves as the critical nexus between circuit design and manufacturing, where its speed and accuracy fundamentally govern the optimization quality of downstream resolution enhancement techniques (RET). While machine learning promises to circumvent computational limitations of lithography process through data-driven or physics-informed approximations of computational lithography, existing simulators suffer from inadequate lithographic awareness due to insufficient training data capturing essential process variations and mask correction rules. We present LithoSim, the most comprehensive lithography simulation benchmark to date, featuring over $4$ million high-resolution input-output pairs with rigorous physical correspondence. The dataset systematically incorporates alterable optical source distributions, metal and via mask topologies with optical proximity correction (OPC) variants, and process windows reflecting fab-realistic variations. By integrating domain-specific metrics spanning AI performance and lithographic fidelity, LithoSim establishes a unified evaluation framework for data-driven and physics-informed computational lithography. The data (https://huggingface.co/datasets/grandiflorum/LithoSim), code (https://dw-hongquan.github.io/LithoSim), and pre-trained models (https://huggingface.co/grandiflorum/LithoSim) are released openly to support the development of hybrid ML-based and high-fidelity lithography simulation for the benefit of semiconductor manufacturing.

Applications · Everything Else

Zitong Lan, Yiduo Hao, Mingmin Zhao

Achieving immersive auditory experiences in virtual environments requires flexible sound modeling that supports dynamic source positions. In this paper, we introduce a task called resounding, which aims to estimate room impulse responses at arbitrary emitter location from a sparse set of measured emitter positions, analogous to the relighting problem in vision. We leverage the reciprocity property and introduce Versa, a physics-inspired approach to facilitating acoustic field learning. Our method creates physically valid samples with dense virtual emitter positions by exchanging emitter and listener poses. We also identify challenges in deploying reciprocity due to emitter/listener gain patterns and propose a self-supervised learning approach to address them. Results show that Versa substantially improve the performance of acoustic field learning on both simulated and real-world datasets across different metrics. Perceptual user studies show that Versa can greatly improve the immersive spatial sound experience.

Applications · Robotics

Yuyang Li, Wenxin Du, Chang Yu, Puhao Li, Zihang Zhao, Tengyu Liu, Chenfanfu Jiang, Yixin Zhu, Siyuan Huang

Tactile sensing is crucial for achieving human-level robotic capabilities in manipulation tasks. As a promising solution, Vision-based Tactile Sensors (VBTSs) offer high spatial resolution and cost-effectiveness, but present unique challenges in robotics for their complex physical characteristics and visual signal processing requirements. The lack of efficient and accurate simulation tools for VBTSs has significantly limited the scale and scope of tactile robotics research. We present Taccel, a high-performance simulation platform that integrates Incremental Potential Contact (IPC) and Affine Body Dynamics (ABD) to model robots, tactile sensors, and objects with both accuracy and unprecedented speed, achieving a total of 915 FPS with 4096 parallel environments. Unlike previous simulators that operate at sub-real-time speeds with limited parallelization, Taccel provides precise physics simulation and realistic tactile signals while supporting flexible robot-sensor configurations through user-friendly APIs. Through extensive validation in object recognition, robotic grasping, and articulated object manipulation, we demonstrate precise simulation and successful sim-to-real transfer. These capabilities position Taccel as a powerful tool for scaling up tactile robotics research and development, potentially transforming how robots interact with and understand their physical environment.

Social Aspects

Xiaoyuan Zhang, Chengdong Ma, Yizhe Huang, Weidong Huang, Siyuan Qi, Song-Chun Zhu, Xue Feng, Yaodong Yang

World models, which explicitly learn environmental dynamics to lay the foundation for planning, reasoning, and decision-making, are rapidly advancing in predicting both physical dynamics and aspects of social behavior, yet predominantly in separate silos. This division results in a systemic failure to model the crucial interplay between physical environments and social constructs, rendering current models fundamentally incapable of adequately addressing the true complexity of real-world systems where physical and social realities are inextricably intertwined. This position paper argues that the systematic, bidirectional unification of physical and social predictive capabilities is the next crucial frontier for world model development. We contend that comprehensive world models must holistically integrate objective physical laws with the subjective, evolving, and context-dependent nature of social dynamics. Such unification is paramount for AI to robustly navigate complex real-world challenges and achieve more generalizable intelligence. This paper substantiates this imperative by analyzing core impediments to integration, proposing foundational guiding principles (ACE Principles), and outlining a conceptual framework alongside a research roadmap towards truly holistic world models.

Social Aspects · Trustworthy Machine Learning

Yichen Wang, Hangtao Zhang, Hewen Pan, Ziqi Zhou, Xianlong Wang, Peijin Guo, Lulu Xue, Shengshan Hu, Minghui Li, Leo Yu Zhang

Vision-Language Models (VLMs), with their strong reasoning and planning capabilities, are widely used in embodied decision-making (EDM) tasks in embodied agents, such as autonomous driving and robotic manipulation. Recent research has increasingly explored adversarial attacks on VLMs to reveal their vulnerabilities. However, these attacks either rely on overly strong assumptions, requiring full knowledge of the victim VLM, which is impractical for attacking VLM-based agents, or exhibit limited effectiveness. The latter stems from disrupting most semantic information in the image, which leads to a misalignment between the perception and the task context defined by system prompts. This inconsistency interrupts the VLM's reasoning process, resulting in invalid outputs that fail to affect interactions in the physical world. To this end, we propose a fine-grained adversarial attack framework, AdvEDM, which modifies the VLM's perception of only a few key objects while preserving the semantics of the remaining regions. This attack effectively reduces conflicts with the task context, making VLMs output valid but incorrect decisions and affecting the actions of agents, thus posing a more substantial safety threat in the physical world. We design two variants of based on this framework, AdvEDM-R and AdvEDM-A, which respectively remove the semantics of a specific object from the image and add the semantics of a new object into the image. The experimental results in both general scenarios and EDM tasks demonstrate fine-grained control and excellent attack performance.

Deep Learning · Everything Else

Sungwoo Park

We introduce Neural Hamiltonian Diffusion, a unified framework for learning stochastic Hamiltonian dynamics on differentiable manifolds. While Hamiltonian Neural Networks (HNNs) model conservative systems in flat Euclidean space, they fail to account for geometric structure and intrinsic stochasticity. Conversely, diffusion models on Riemannian manifolds offer geometry-aware stochastic modeling but lack physical inductive biases. Our method parameterizes a Hamiltonian with a neural network and defines its dynamics as a stochastic differential equation on a (pseudo-)Riemannian manifold equipped with a Poisson structure. This enables physically consistent modeling of dynamics on curved, periodic, or causally structured spaces. We demonstrate that the proposed geometric dynamics generalizes existing approaches and applies to systems ranging from molecular dynamics to relativistic n-body problems.

Applications · Physics

Xihang Yue, Yi Yang, Linchao Zhu

The limited availability of high-quality training data poses a major obstacle in data-driven PDE solving, where expensive data collection and resolution constraints severely impact the ability of neural operator networks to learn and generalize the underlying physical system. To address this challenge, we propose DeltaPhi, a novel learning framework that transforms the PDE solving task from learning direct input-output mappings to learning the residuals between similar physical states, a fundamentally different approach to neural operator learning. This reformulation provides implicit data augmentation by exploiting the inherent stability of physical systems where closer initial states lead to closer evolution trajectories. DeltaPhi is architecture-agnostic and can be seamlessly integrated with existing neural operators to enhance their performance. Extensive experiments demonstrate consistent and significant improvements across diverse physical systems including regular and irregular domains, different neural architectures, multiple training data amount, and cross-resolution scenarios, confirming its effectiveness as a general enhancement for neural operators in data-limited PDE solving.

Computer Vision · 3D Rendering & Reconstruction

Jinkun Hao, Naifu Liang, Zhen Luo, Xudong XU, Weipeng Zhong, Ran Yi, Yichen Jin, Zhaoyang Lyu, Feng Zheng, Lizhuang Ma 等

The ability of robots to interpret human instructions and execute manipulation tasks necessitates the availability of task-relevant tabletop scenes for training. However, traditional methods for creating these scenes rely on time-consuming manual layout design or purely randomized layouts, which are limited in terms of plausibility or alignment with the tasks. In this paper, we formulate a novel task, namely task-oriented tabletop scene generation, which poses significant challenges due to the substantial gap between high-level task instructions and the tabletop scenes. To support research on such a challenging task, we introduce \textbf{MesaTask-10K}, a large-scale dataset comprising approximately 10,700 synthetic tabletop scenes with \emph{manually crafted layouts} that ensure realistic layouts and intricate inter-object relations. To bridge the gap between tasks and scenes, we propose a \textbf{Spatial Reasoning Chain} that decomposes the generation process into object inference, spatial interrelation reasoning, and scene graph construction for the final 3D layout. We present \textbf{MesaTask}, an LLM-based framework that utilizes this reasoning chain and is further enhanced with DPO algorithms to generate physically plausible tabletop scenes that align well with given task descriptions. Exhaustive experiments demonstrate the superior performance of MesaTask compared to baselines in generating task-conforming tabletop scenes with realistic layouts.

Optimization · Neural Network Optimization

Dmitriy Shopkhoev, Ammar Ali, Magauiya Zhussip, Valentin Malykh, Stamatios Lefkimmiatis, Nikos Komodakis, Sergey Zagoruyko

We introduce ReplaceMe, a generalized training-free depth pruning method that effectively replaces transformer blocks with a linear operation, while maintaining high performance for low compression ratios. In contrast to conventional pruning approaches that require additional training or fine-tuning, our approach requires only a small calibration dataset that is used to estimate a linear transformation, which approximates the pruned blocks. The estimated linear mapping can be seam- lessly merged with the remaining transformer blocks, eliminating the need for any additional network parameters. Our experiments show that ReplaceMe consistently outperforms other training-free approaches and remains highly competitive with state-of-the-art pruning methods that involve extensive retraining/fine-tuning and architectural modifications. Applied to several large language models (LLMs), ReplaceMe achieves up to 25% pruning while retaining approximately 90% of the original model’s performance on open benchmarks—without any training or healing steps, resulting in minimal computational overhead. We provide an open- source library implementing ReplaceMe alongside several state-of-the-art depth pruning techniques, available at https://github.com/mts-ai/ReplaceMe.

Optimization · Large Scale, Parallel and Distributed

Guoliang He, Youhe Jiang, Wencong Xiao, Jiang Kaihua, Shuguang Wang, Jun Wang, Du Zixian, Zhuo Jiang, Xinlei Zhang, Binhang Yuan 等

The scaling law for large language models (LLMs) depicts that the path towards machine intelligence necessitates training at large scale. Thus, companies continuously build large-scale GPU clusters, and launch training jobs that span over thousands of computing nodes. However, LLM pre-training presents unique challenges due to its complex communication patterns, where GPUs exchange data in sparse yet high-volume bursts within specific groups. Inefficient resource scheduling exacerbates bandwidth contention, leading to suboptimal training performance. This paper presents Arnold, a scheduling system summarizing our experience to effectively align LLM communication patterns to data center topology at scale. In-depth characteristic study is performed to identify the impact of physical network topology to LLM pre-training jobs. Based on the insights, we develop a scheduling algorithm to effectively align communication patterns to physical network topology in data centers. Through simulation experiments, we show the effectiveness of our algorithm in reducing the maximum spread of communication groups by up to $1.67$x. In production training, our scheduling system improves the end-to-end performance by $10.6\%$ when training with more than $9600$ Hopper GPUs, a significant improvement for our training pipeline.

Computer Vision · Image and Video Generation

Xiangdong Zhang, Jiaqi Liao, Shaofeng Zhang, Fanqing Meng, Xiangpeng Wan, Junchi Yan, Yu Cheng

Recent advancements in text-to-video (T2V) diffusion models have enabled high-fidelity and realistic video synthesis. However, current T2V models often struggle to generate physically plausible content due to their limited inherent ability to accurately understand physics. We found that while the representations within T2V models possess some capacity for physics understanding, they lag significantly behind those from recent video self-supervised learning methods. To this end, we propose a novel framework called {VideoREPA}, which distills physics understanding capability from video understanding foundation models into T2V models by aligning token-level relations. This closes the physics understanding gap and enables more physics-plausible generation. Specifically, we introduce the {Token Relation Distillation (TRD) loss}, leveraging spatio-temporal alignment to provide soft guidance suitable for finetuning powerful pre-trained T2V models—a critical departure from prior representation alignment (REPA) methods. To our knowledge, VideoREPA is the first REPA method designed for finetuning T2V models and specifically for injecting physical knowledge. Empirical evaluations show that VideoREPA substantially enhances the physics commonsense of baseline method, CogVideoX, achieving significant improvement on relevant benchmarks and demonstrating a strong capacity for generating videos consistent with intuitive physics. Code and more video results are available at https://videorepa.github.io/.

Applications · Physics

Jiachen Yao, Abbas Mammadov, Julius Berner, Gavin Kerrigan, Jong Chul Ye, Kamyar Azizzadenesheli, Animashree Anandkumar

We propose a general framework for conditional sampling in PDE-based inverse problems, targeting the recovery of whole solutions from extremely sparse or noisy measurements. This is accomplished by a function-space diffusion model and plug-and-play guidance for conditioning. Our method first trains an unconditional discretization-agnostic denoising model using neural operator architectures. At inference, we refine the samples to satisfy sparse observation data via a gradient-based guidance mechanism. Through rigorous mathematical analysis, we extend Tweedie's formula to infinite-dimensional Hilbert spaces, providing the theoretical foundation for our posterior sampling approach. Our method (FunDPS) accurately captures posterior distribution in function spaces under minimal supervision and severe data scarcity. Across five PDE tasks with only 3\% observation, our method achieves an average 32\% accuracy improvement over state-of-the-art fixed-resolution diffusion baselines while reducing sampling steps by 4x. Furthermore, multi-resolution fine-tuning ensures strong cross-resolution generalizability and speedup. To the best of our knowledge, this is the first diffusion-based framework to operate independently of discretization, offering a practical and flexible solution for forward and inverse problems in the context of PDEs. Code is available at https://github.com/neuraloperator/FunDPS.