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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.

Juhyeon Kim, Craig Benko, Magnus Wrenninge, Ryusuke Villemin, Zeb W. Barber, Wojciech Jarosz, Adithya Pediredla

Optical heterodyne detection (OHD) employs coherent light and optical interference techniques (Fig. 1-(A)) to extract physical parameters, such as velocity or distance, which are encoded in the frequency modulation of the light. With its superior signal-to-noise ratio compared to incoherent detection methods, such as time-of-flight lidar, OHD has become integral to applications requiring high sensitivity, including autonomous navigation, atmospheric sensing, and biomedical velocimetry. However, current simulation tools for OHD focus narrowly on specific applications, relying on domain-specific settings like restricted reflection functions, scene configurations, or single-bounce assumptions, which limit their applicability. In this work, we introduce a flexible and general framework for spectral-domain simulation of OHD. We demonstrate that classical radiometry-based path integral formulation can be adapted and extended to simulate the OHD measurements in the spectral domain. This enables us to leverage the rich modeling and sampling capabilities of existing Monte Carlo path tracing techniques. Our formulation shares structural similarities with transient rendering but operates in the spectral domain and accounts for the Doppler effect (Fig. 1-(B)). While simulators for the Doppler effect in incoherent (intensity) detection methods exist, they are largely not suitable to simulate OHD. We use a microsurface interpretation to show that these two Doppler imaging techniques capture different physical quantities and thus need different simulation frameworks. We validate the correctness and predictive power of our simulation framework by qualitatively comparing the simulations with real-world captured data for three different OHD applications—FMCW lidar, blood flow velocimetry, and wind Doppler lidar (Fig. 1-(C)).

Paul Himmler, Tobias Günther

In recent years, grid-free Monte Carlo methods have gained increasing popularity for solving fundamental partial differential equations. For a given point in the domain, the Walk-on-Spheres method solves a boundary integral equation by integrating recursively over the largest possible sphere. When the walks approach boundaries with Dirichlet conditions, the number of path vertices increases considerably, since the step size becomes smaller with decreasing distance to the boundary. In practice, the walks are terminated once they reach an epsilon-shell around the boundary. This, however, introduces bias, leading to a trade-off between accuracy and performance. Instead of using spheres, we propose to utilize geometric primitives that share more than one point with the boundary to increase the likelihood of immediately terminating. Along the boundary of those new geometric primitives a sampling probability is needed, which corresponds to the exit probability of a Brownian motion. This is known as a first passage problem. Utilizing that Laplace equations are invariant under conformal maps, we transform exit points from unit circles to the exit points of our geometric primitives, for which we describe a suitable placement strategy. With this, we obtain a novel approach to solve the Laplace equation in two dimensions, which does not require an epsilon-shell, significantly reduces the number of path vertices, and reduces inaccuracies near Dirichlet boundaries.

Pedro Figueirêdo, Qihao He, Steve Bako, Nima Khademi Kalantari

We propose a neural approach for estimating spatially varying light selection distributions to improve importance sampling in Monte Carlo rendering, particularly for complex scenes with many light sources. Our method uses a neural network to predict the light selection distribution at each shading point based on local information, trained by minimizing the KL-divergence between the learned and target distributions in an online manner. To efficiently manage hundreds or thousands of lights, we integrate our neural approach with light hierarchy techniques, where the network predicts cluster-level distributions and existing methods sample lights within clusters. Additionally, we introduce a residual learning strategy that leverages initial distributions from existing techniques, accelerating convergence during training. Our method achieves superior performance across diverse and challenging scenes.

Jiawei Huang 0005, Shaokun Zheng, Kun Xu 0003, Yoshifumi Kitamura, Jiaping Wang

We introduce a guided lens sampling method for efficient rendering of circles of confusion (CoCs). While traditional Monte Carlo techniques simulate depth-of-field (DoF) effects by perturbing camera rays at the lens, uniform lens sampling often results in significant noise by failing to prioritize rays toward highlight regions in the scene. Although path guiding has proven effective for global illumination by learning importance distributions for incoming radiance, no comparable guiding technique for CoCs exists, primarily due to the strong parallax between adjacent pixels. We model highlight spots in world space using a globally shared radiance field, which is then transformed into lens space through a bipolar-cone projection to guide camera ray generation. We implement this theory using 3D Gaussians, achieving fast, robust guiding with minimal computational and storage overhead, making it suitable for production rendering. We also propose two extensions to further enhance local adaptation. Our experiments show that this approach significantly improves the sampling efficiency for CoC rendering.

Tianyu Huang, Jingwang Ling, Shuang Zhao, Feng Xu 0005

Walk on stars (WoSt) has shown its power in being applied to Monte Carlo methods for solving partial differential equations, but the sampling techniques in WoSt are not satisfactory, leading to high variance. We propose a guiding-based importance sampling method to reduce the variance of WoSt. Drawing inspiration from path guiding in rendering, we approximate the directional distribution of the recursive term of WoSt using online-learned parametric mixture distributions, decoded by a lightweight neural field. This adaptive approach enables importance sampling the recursive term, which lacks shape information before computation. We introduce a reflection technique to represent guiding distributions at Neumann boundaries and incorporate multiple importance sampling with learnable selection probabilities to further reduce variance. We also present a practical GPU implementation of our method. Experiments show that our method effectively reduces variance compared to the original WoSt, given the same time or the same sample budget. Code and data for this paper are at https://github.com/tyanyuy3125/elaina.

Shakiba Kheradmand, Delio Vicini, George Kopanas, Dmitry Lagun, Kwang Moo Yi, Mark Matthews, Andrea Tagliasacchi

3D Gaussian splatting (3DGS) is a popular radiance field method, with many application-specific extensions. Most variants rely on the same core algorithm: depth-sorting of Gaussian splats then rasterizing in primitive order. This ensures correct alpha compositing, but can cause rendering artifacts due to built-in approximations. Moreover, for a fixed representation, sorted rendering offers little control over render cost and visual fidelity. For example, and counter-intuitively, rendering a lower-resolution image is not necessarily faster. In this work, we address the above limitations by combining 3D Gaussian splatting with stochastic rasterization. Concretely, we leverage an unbiased Monte Carlo estimator of the volume rendering equation. This removes the need for sorting, and allows for accurate 3D blending of overlapping Gaussians. The number of Monte Carlo samples further imbues 3DGS with a way to trade off computation time and quality. We implement our method using OpenGL shaders, enabling efficient rendering on modern GPU hardware. At a reasonable visual quality, our method renders more than four times faster than sorted rasterization.

Fangyikang Wang, Hubery Yin, Lei Qian, Yinan Li, Shaobin Zhuang, Huminhao Zhu, Yilin Zhang, Yanlong Tang, Chao Zhang, Hanbin Zhao 等

The emerging diffusion models (DMs) have demonstrated the remarkable capability of generating images via learning the noised score function of the data distribution. Current DM sampling techniques typically rely on first-order Langevin dynamics at each noise level, with efforts concentrated on refining inter-level denoising strategies. While leveraging additional second-order Hessian geometry to enhance the sampling quality of Langevin is a common practice in Markov chain Monte Carlo (MCMC), the naive attempts to utilize Hessian geometry in high-dimensional DMs lead to quadratic-complexity computational costs, rendering them non-scalable. In this work, we introduce a novel Levenberg-Marquardt-Langevin (LML) method that approximates the diffusion Hessian geometry in a training-free manner, drawing inspiration from the celebrated Levenberg-Marquardt optimization algorithm. Our approach introduces two key innovations: (1) A low-rank approximation of the diffusion Hessian, leveraging the DMs' inherent structure and circumventing explicit quadratic-complexity computations; (2) A damping mechanism to stabilize the approximated Hessian. This LML approximated Hessian geometry enables the diffusion sampling to execute more accurate steps and improve the image generation quality. We further conduct a theoretical analysis to substantiate the approximation error bound of low-rank approximation and the convergence property of the damping mechanism. Extensive experiments across multiple pretrained DMs validate that the LML method significantly improves image generation quality, with negligible computational overhead.

Pengcheng Zheng, Kecheng Chen, Jiaxin Huang 0006, Bohao Chen, Ju Liu, Yazhou Ren 0001, Xiaorong Pu

Medical image restoration tasks aim to recover high-quality images from degraded observations, exhibiting emergent desires in many clinical scenarios, such as low-dose CT image denoising, MRI super-resolution, and MRI artifact removal. Despite the success achieved by existing deep learning-based restoration methods with sophisticated modules, they struggle with rendering computationally-efficient reconstruction results. Moreover, they usually ignore the reliability of the restoration results, which is much more urgent in medical systems. To alleviate these issues, we present LRformer, a Lightweight Transformer-based method via Reliability-guided learning in the frequency domain. Specifically, inspired by the uncertainty quantification in Bayesian neural networks (BNNs), we develop a Reliable Lesion-Semantic Prior Producer (RLPP). RLPP leverages Monte Carlo (MC) estimators with stochastic sampling operations to generate sufficiently-reliable priors by performing multiple inferences on the foundational medical image segmentation model, MedSAM. Additionally, instead of directly incorporating the priors in the spatial domain, we decompose the cross-attention (CA) mechanism into real symmetric and imaginary anti-symmetric parts via fast Fourier transform (FFT), resulting in the design of the Guided Frequency Cross-Attention (GFCA) solver. By leveraging the conjugated symmetric property of FFT, GFCA reduces the computational complexity of naive CA by nearly half. Extensive experimental results in various tasks demonstrate the superiority of the proposed LRformer in both effectiveness and efficiency.

Bo Peng, Jie Lu 0001, Yonggang Zhang 0003, Guangquan Zhang 0001, Zhen Fang 0001

Out-of-distribution (OOD) detection has emerged as a pivotal approach for enhancing the reliability of machine learning models, considering the potential for test data to be sampled from classes disparate from in-distribution (ID) data employed during model training. Detecting those OOD data is typically realized as a distance measurement problem, where those deviating far away from the training distribution in the learned feature space are considered OOD samples. Advanced works have shown great success in learning with prototypes for feature-based OOD detection methods, where each ID class is represented with single or multiple prototypes. However, modeling with a finite number of prototypes would fail to maximally capture intra-class variations. In view of this, this paper extends the existing prototype-based learning paradigm to an infinite setting. This motivates us to design two feasible formulations for the Distributional Prototype Learning (DPL) objective, where, to avoid intractable computation and exploding parameters caused by the infinity nature, our key idea is to model an infinite number of discrete prototypes of each ID class with a class-wise continuous distribution. We theoretically analyze both alternatives, identifying the more stable-converging version of the learning objective. We show that, by sampling prototypes from a mixture of class-conditioned Gaussian distributions, the objective can be efficiently computed in a closed form without resorting to the computationally expensive Monte-Carlo approximation of the involved expectation terms. Extensive evaluations across mainstream OOD detection benchmarks empirically manifest that our proposed DPL has established a new state-of-the-art in various OOD settings.

Haidong Kang, Lianbo Ma, Guo Yu, Shangce Gao

Mixed precision quantization (MPQ) is an effective quantization approach to achieve accuracy-complexity trade-off of neural network, through assigning different bit-widths to network activations and weights in each layer. The typical way of existing MPQ methods is to optimize quantization policies (i.e., bit-width allocation) in a gradient descent manner, termed as Differentiable MPQ (DMPQ). At the end of the search, the bit-width associated to the quantization parameters which has the largest value will be selected to form the final mixed precision quantization policy, with the implicit assumption that the values of quantization parameters reflect the operation contribution to the accuracy improvement. While much has been discussed about the MPQ’s improvement, the bit-width selection process has received little attention. We study this problem and argue that the magnitude of quantization parameters does not necessarily reflect the actual contribution of the bit-width to the task performance. Then, we propose a Shapley-based MPQ (SMPQ) method, which measures the bit-width operation’s direct contribution on the MPQ task. To reduce computation cost, a Monte Carlo sampling-based approximation strategy is proposed for Shapley computation. Extensive experiments on mainstream benchmarks demonstrate that our SMPQ consistently achieves state-of-the-art performance than gradient-based competitors.

Zun Li, Marc Lanctot, Kevin R. McKee, Luke Marris, Ian Gemp, Daniel Hennes, Paul Muller, Kate Larson, Yoram Bachrach, Michael P. Wellman

Opponent modeling methods typically involve two crucial steps: building a belief distribution over opponents' strategies, and exploiting this opponent model by playing a best response. However, existing approaches typically require domain-specific heurstics to come up with such a model, and algorithms for approximating best responses are hard to scale in large, imperfect information domains. In this work, we introduce a scalable and generic multiagent training regime for opponent modeling using deep game-theoretic reinforcement learning. We first propose Generative Best Respoonse (GenBR), a best response algorithm based on Monte-Carlo Tree Search (MCTS) with a learned deep generative model that samples world states during planning. This new method scales to large imperfect information domains and can be plug and play in a variety of multiagent algorithms. We use this new method under the framework of Policy Space Response Oracles (PSRO), to automate the generation of an offline opponent model via iterative game-theoretic reasoning and population-based training. We propose using solution concepts based on bargaining theory to build up an opponent mixture, which we find identifying profiles that are near the Pareto frontier. Then GenBR keeps updating an online opponent model and reacts against it during gameplay. We conduct behavioral studies where human participants negotiate with our agents in Deal-or-No-Deal, a class of bilateral bargaining games. Search with generative modeling finds stronger policies during both training time and test time, enables online Bayesian co-player prediction, and can produce agents that achieve comparable social welfare and Nash bargaining score negotiating with humans as humans trading among themselves.

Jianing Wang, Yang Zhou, Xiaocheng Zhang, Mengjiao Bao, Peng Yan

Iterative preference optimization has recently become one of the de-facto training paradigms for large language models (LLMs), but the performance is still underwhelming due to too much noisy preference data yielded in the loop. To combat this issue, we present an Uncertainty-enhanced Preference Optimization (UPO) framework to make the LLM self-evolve with reliable feedback. The key idea is mitigating the noisy preference pairs derived from the current policy and reward models by performing pair-wise uncertainty estimation and judiciously reliable feedback sampling. To reach this goal, we thus introduce an estimator model, which incorporates Monte Carlo (MC) dropout in Bayesian neural network (BNN) to perform uncertainty estimation for the batch of preference pairs. Compared to the existing methods that directly filter generated responses based on the reward score, the estimator focuses on the model uncertainty in a pair-wise manner and effectively bypasses the confirmation bias problem of the reward model. Additionally, we also propose an uncertainty-enhanced self-evolution algorithm to better improve the LLM robustly align with these reliable feedback data. Extensive experiments over multiple benchmarks demonstrate our framework substantially improves the performance of iterative preference optimization.

Liam Anthony Kruse, Alexandros Tzikas, Harrison Delecki, Mansur Arief, Mykel J. Kochenderfer

Importance sampling is a rare event simulation technique used in Monte Carlo simulations to bias the sampling distribution towards the rare event of interest. By assigning appropriate weights to sampled points, importance sampling allows for more efficient estimation of rare events or tails of distributions. However, importance sampling can fail when the proposal distribution does not effectively cover the target distribution. In this work, we propose a method for more efficient sampling by updating the proposal distribution in the latent space of a normalizing flow. Normalizing flows learn an invertible mapping from a target distribution to a simpler latent distribution. The latent space can be more easily explored during the search for a proposal distribution, and samples from the proposal distribution are recovered in the space of the target distribution via the invertible mapping. We empirically validate our methodology on simulated robotics applications such as autonomous racing and aircraft ground collision avoidance.

Shiqi Gao, Tianxiang Gong, Zijie Lin, Runhua Xu, Haoyi Zhou, Jianxin Li

Uncertainty estimation is essential for practical applications such as decision-making, risk assessment, and human-AI collaboration. However, Uncertainty estimation in open-ended question-answering (QA) tasks presents unique challenges. The output space for open-ended QA is vast and discrete, and the autoregressive nature of LLMs, combined with the rapid increase in model parameters, makes inference sampling significantly costly. An ideal uncertainty estimation for LLMs should meet two criteria: 1) incur no additional inference cost and 2) capture the semantic dependencies of token-level uncertainty within sequences. We propose a promising solution that converts redundancy into randomness in the extensive parameters of LLMs to quantify knowledge uncertainty. We can obtain token-level Monte Carlo samples without multiple inferences by introducing randomness during a single forward pass. We theoretically analyze the FLUE sampling method and employ a post-processing method to learn the state transitions from token uncertainty to sequence uncertainty. In open-ended question-answering tasks, we demonstrate that FLUE can achieve competitive performance in estimating the uncertainty of generated sentences without adding extra inference overhead.

Mateusz Gajewski, Adam Karczmarz, Mateusz Rapicki, Piotr Sankowski

In this paper, we consider a perturbation-based metric of predictive faithfulness of feature rankings (or attributions) that we call PGI squared When applied to decision tree-based regression models, the metric can be computed exactly and efficiently for arbitrary independent feature perturbation distributions. In particular, the computation does not involve Monte Carlo sampling that has been typically used for computing similar metrics and which is inherently prone to inaccuracies. As a second contribution, we proposed a procedure for constructing feature ranking based on PGI squared. Our results indicate the proposed ranking method is comparable to the widely recognized SHAP explainer, offering a viable alternative for assessing feature importance in tree-based models.

Chun Gu, Xiaofei Wei, Li Zhang, Xiatian Zhu

Inverse rendering aims to recover scene geometry, material properties, and lighting from multi-view images. Given the complexity of light-surface interactions, importance sampling is essential for the evaluation of the rendering equation, as it reduces variance and enhances the efficiency of Monte Carlo sampling. Existing inverse rendering methods typically use pre-defined non-learnable importance samplers in prior manually, struggling to effectively match the spatially and directionally varied integrand and resulting in high variance and suboptimal performance. To address this limitation, we propose the concept of learning a spatially and directionally aware importance sampler for the rendering equation to accurately and flexibly capture the unconstrained complexity of a typical scene. We further formulate TensoFlow, a generic approach for sampler learning in inverse rendering, enabling to closely match the integrand of the rendering equation spatially and directionally. Concretely, our sampler is parameterized by normalizing flows, allowing both directional sampling of incident light and probability density function (PDF) inference. To capture the characteristics of the sampler spatially, we learn a tensorial representation of the scene space, which imposes spatial conditions, together with reflected direction, leading to spatially and directionally aware sampling distributions. Our model can be optimized by minimizing the difference between the integrand and our normalizing flow. Extensive experiments validate the superiority of TensoFlow over prior alternatives on both synthetic and real-world benchmarks.

Chun Gu, Xiaofei Wei, Zixuan Zeng, Yuxuan Yao, Li Zhang

In inverse rendering, accurately modeling visibility and indirect radiance for incident light is essential for capturing secondary effects. Due to the absence of a powerful Gaussian ray tracer, previous 3DGS-based methods have either adopted a simplified rendering equation or used learnable parameters to approximate incident light, resulting in inaccurate material and lighting estimations. To this end, we introduce the inter-reflective Gaussian splatting (IRGS) framework for inverse rendering. To capture inter-reflection, we apply the full rendering equation without simplification and compute incident radiance on the fly using the proposed differentiable 2D Gaussian ray tracing. Additionally, we present an efficient optimization scheme to handle the computational demands of Monte Carlo sampling for rendering equation evaluation. Furthermore, we introduce a novel strategy for querying the indirect radiance of incident light when relighting the optimized scenes. Extensive experiments on multiple standard benchmarks validate the effectiveness of IRGS, demonstrating its capability to accurately model complex inter-reflection effects.

Toan Nguyen, Kien Do, Duc Kieu, Thin Nguyen

We introduce a theoretical framework for diffusion-based image editing by formulating it as a reverse-time bridge modeling problem. This approach modifies the backward process of a pretrained diffusion model to construct a bridge that converges to an implicit distribution associated with the editing target at time 0. Building on this framework, we propose h-Edit, a novel editing method that utilizes Doob's h-transform and Langevin Monte Carlo to decompose the update of an intermediate edited sample into two components: a "reconstruction" term and an "editing" term. This decomposition provides flexibility, allowing the reconstruction term to be computed via existing inversion techniques and enabling the combination of multiple editing terms to handle complex editing tasks. To our knowledge, h-Edit is the first training-free method capable of performing simultaneous text-guided and reward-model-based editing. Extensive experiments, both quantitative and qualitative, show that h-Edit outperforms state-of-the-art baselines in terms of editing effectiveness and faithfulness.

Tamás Ficsor, Gábor Berend

The growing deployment of deep learning models in real-world applications necessitates not only high predictive accuracy, but also mechanism to identify unreliable predictions, especially in high-stakes scenarios where decision risk must be minimized. Existing methods estimate uncertainty by leveraging predictive confidence (e.g., Softmax Response), structural characteristics of representation space (e.g., Mahalanobis distance), or stochastic variation in model outputs (e.g., Bayesian inference techniques such as Monte Carlo Dropout). In this work, we propose a novel uncertainty estimation (UE) framework based on sparse dictionary learning by identifying dictionary atoms associated with misclassified samples. We leverage pointwise mutual information (PMI) to quantify the association between sparse features and predictive failure. Our method – Sparsity-based Uncertainty Estimation (SUE) – is computationally efficient, offers interpretability via atom-level analysis of the dictionary, has no assumption about the class distribution (unlike Mahalanobis distance). We evaluated SUE on several NLU benchmarks (GLUE and ANLI tasks) and sentiment analysis benchmarks (Twitter, ParaDetox, and Jigsaw). In general, SUE outperforms or matches the performance of other methods. SUE performs particularly well when there is considerable uncertainty in the model, i.e., when the model lacks high precision.