Physics-Informed ML

Recent progress on Remote Sensing Foundation Models (RSFMs) aims toward universal representations for Earth observation imagery. However, current efforts often scale up in size significantly without addressing efficiency constraints critical for real-world applications (e.g., onboard processing, rapid disaster response) or treat multispectral (MS) data as generic imagery, overlooking valuable physical priors. We introduce PhySwin, a foundation model for MS data that integrates physical priors with computational efficiency. PhySwin combines three innovations: (i) physics-informed pretraining objectives leveraging radiometric constraints to enhance feature learning; (ii) an efficient MixMAE formulation tailored to SwinV2 for low-FLOP, scalable pretraining; and (iii) token-efficient spectral embedding to retain spectral detail without increasing token counts. Pretrained on over 1M Sentinel-2 tiles, PhySwin achieves SOTA results (+1.32\% mIoU segmentation, +0.80\% F1 change detection) while reducing inference latency by up to 14.4$\times$ and computational complexity by up to 43.6$\times$ compared to ViT-based RSFMs.

The physics-informed neural networks (PINNs) are widely applied in solving differential equations. However, few studies have discussed their consistency. In this paper, we consider the consistency of PINNs when applied to second-order elliptic equations with Dirichlet boundary conditions. We first provide the necessary and sufficient condition for the consistency of the physics-informed kernel gradient flow algorithm, and then as a direct corollary, when the neural network is sufficiently wide, we obtain a necessary and sufficient condition for the consistency of PINNs based on the neural tangent kernel theory. We also estimate the non-asymptotic loss bounds of physics-informed kernel gradient flow and PINN under suitable stronger assumptions. Finally, these results inspires us to construct a notable pathological example where the PINN method is inconsistent.

Physics sensing plays a central role in many scientific and engineering domains, which inherently involves two coupled tasks: reconstructing dense physical fields from sparse observations and optimizing scattered sensor placements to observe maximum information. While deep learning has made rapid advances in sparse-data reconstruction, existing methods generally omit optimization of sensor placements, leaving the mutual enhancement between reconstruction and placement on the shelf. To change this suboptimal practice, we propose PhySense, a synergistic two-stage framework that learns to jointly reconstruct physical fields and to optimize sensor placements, both aiming for accurate physics sensing. The first stage involves a flow-based generative model enhanced by cross-attention to adaptively fuse sparse observations. Leveraging the reconstruction feedback, the second stage performs sensor placement via projected gradient descent to satisfy spatial constraints. We further prove that the learning objectives of the two stages are consistent with classical variance-minimization principles, providing theoretical guarantees. Extensive experiments across three challenging benchmarks, especially a 3D geometry dataset, indicate PhySense achieves state-of-the-art physics sensing accuracy and discovers informative sensor placements previously unconsidered. Code is available at this repository: https://github.com/thuml/PhySense.

Reinforcement learning (RL) in continuous state-action spaces remains challenging in scientific computing due to poor sample efficiency and lack of pathwise physical consistency. We introduce Differential Reinforcement Learning (Differential RL), a novel framework that reformulates RL from a continuous-time control perspective via a differential dual formulation. This induces a Hamiltonian structure that embeds physics priors and ensures consistent trajectories without requiring explicit constraints. To implement Differential RL, we develop Differential Policy Optimization (dfPO), a pointwise, stage-wise algorithm that refines local movement operators along the trajectory for improved sample efficiency and dynamic alignment. We establish pointwise convergence guarantees, a property not available in standard RL, and derive a competitive theoretical regret bound of $\mathcal{O}(K^{5/6})$. Empirically, dfPO outperforms standard RL baselines on representative scientific computing tasks, including surface modeling, grid control, and molecular dynamics, under low-data and physics-constrained conditions.

Recent advances in Scientific Machine Learning have shown that second-order methods can enhance the training of Physics-Informed Neural Networks (PINNs), making them a suitable alternative to traditional numerical methods for Partial Differential Equations (PDEs). However, second-order methods induce large memory requirements, making them scale poorly with the model size. In this paper, we define a local Mixture of Experts (MoE) combining the parameter-efficiency of ensemble models and sparse coding to enable the use of second-order training. Our model -- PINN Balls -- also features a fully learnable domain decomposition structure, achieved through the use of Adversarial Adaptive Sampling (AAS), which adapts the DD to the PDE and its domain. PINN Balls achieves better accuracy than the state-of-the-art in scientific machine learning, while maintaining invaluable scalability properties and drawing from a sound theoretical background.

Variational quantum Eigensolver (VQE) is a leading candidate for harnessing quantum computers to advance quantum chemistry and materials simulations, yet its training efficiency deteriorates rapidly for large Hamiltonians. Two issues underlie this bottleneck: (i) the no-cloning theorem imposes a linear growth in circuit evaluations with the number of parameters per gradient step; and (ii) deeper circuits encounter barren plateaus (BPs), leading to exponentially increasing measurement overheads. To address these challenges, here we propose a deep learning framework, dubbed Titan, which identifies and freezes inactive parameters of a given ansätze at initialization for a specific class of Hamiltonians, reducing the optimization overhead without sacrificing accuracy. The motivation of Titan starts with our empirical findings that a subset of parameters consistently has negligible influence on training dynamics. Its design combines a theoretically grounded data construction strategy, ensuring each training example is informative and BP-resilient, with an adaptive neural architecture that generalizes across ansätze of varying sizes. Across benchmark transverse-field Ising models, Heisenberg models, and multiple molecule systems up to $30$ qubits, Titan achieves up to $3\times$ faster convergence and $40$–$60\%$ fewer circuit evaluations than state-of-the-art baselines, while matching or surpassing their estimation accuracy. By proactively trimming parameter space, Titan lowers hardware demands and offers a scalable path toward utilizing VQE to advance practical quantum chemistry and materials science.

World models have become indispensable tools for embodied intelligence, serving as powerful simulators capable of generating realistic robotic videos while addressing critical data scarcity challenges. However, current embodied world models exhibit limited physical awareness, particularly in modeling 3D geometry and motion dynamics, resulting in unrealistic video generation for contact-rich robotic scenarios. In this paper, we present RoboScape, a unified physics-informed world model that jointly learns RGB video generation and physics knowledge within an integrated framework. We introduce two key physics-informed joint training tasks: temporal depth prediction that enhances 3D geometric consistency in video rendering, and keypoint dynamics learning that implicitly encodes physical properties (e.g., object shape and material characteristics) while improving complex motion modeling. Extensive experiments demonstrate that RoboScape generates videos with superior visual fidelity and physical plausibility across diverse robotic scenarios. We further validate its practical utility through downstream applications including robotic policy training with generated data and policy evaluation. Our work provides new insights for building efficient physics-informed world models to advance embodied intelligence research. Our code and demos are available at: https://github.com/tsinghua-fib-lab/RoboScape.

Variational inference often struggles with the posterior geometry exhibited by complex hierarchical Bayesian models. Recent advances in flow‐based variational families and Variationally Inferred Parameters (VIP) each address aspects of this challenge, but their formal relationship is unexplored. Here, we prove that the combination of VIP and a full-rank Gaussian can be represented exactly as a forward autoregressive flow augmented with a translation term and input from the model’s prior. Guided by this theoretical insight, we introduce the Model‐Informed Flow (MIF) architecture, which adds the necessary translation mechanism, prior information, and hierarchical ordering. Empirically, MIF delivers tighter posterior approximations and matches or exceeds state‐of‐the‐art performance across a suite of hierarchical and non‐hierarchical benchmarks.

Evaluating the scientific discovery capabilities of large language model based agents, particularly how they cope with varying environmental complexity and utilize prior knowledge, requires specialized benchmarks currently lacking in the landscape. To address this gap, we introduce PhysGym, a novel benchmark suite and simulation platform for rigorously assessing LLM-based scientific reasoning in interactive physics environments. PhysGym's primary contribution lies in its sophisticated control over the level of prior knowledge provided to the agent. This allows researchers to dissect agent performance along axes including the complexity of the problem and the prior knowledge levels. The benchmark comprises a suite of interactive simulations, where agents must actively probe environments, gather data sequentially under constraints and formulate hypotheses about underlying physical laws. PhysGym provides standardized evaluation protocols and metrics for assessing hypothesis accuracy and model fidelity. We demonstrate the benchmark's utility by presenting results from baseline LLMs, showcasing its ability to differentiate capabilities based on varying priors and task complexity.

Medical image reconstruction from measurement data is a vital but challenging inverse problem. Deep learning approaches have achieved promising results, but often requires paired measurement and high-quality images, which is typically simulated through a forward model, i.e., retrospective reconstruction. However, training on simulated pairs commonly leads to performance degradation on real prospective data due to the retrospective-to-prospective gap caused by incomplete imaging knowledge in simulation. To address this challenge, this paper introduces imaging Knowledge-Informed Dynamic Optimal Transport (KIDOT), a novel dynamic optimal transport framework with optimality in the sense of preserving consistency with imaging physics in transport, that conceptualizes reconstruction as finding a dynamic transport path. KIDOT learns from unpaired data by modeling reconstruction as a continuous evolution path from measurements to images, guided by an imaging knowledge-informed cost function and transport equation. This dynamic and knowledge-aware approach enhances robustness and better leverages unpaired data while respecting acquisition physics. Theoretically, we demonstrate that KIDOT naturally generalizes dynamic optimal transport, ensuring its mathematical rationale and solution existence. Extensive experiments on MRI and CT reconstruction demonstrate KIDOT's superior performance. Code is available at https://github.com/TaoranZheng717/KIDOT.

Over the past decades, pansharpening has contributed greatly to numerous remote sensing applications, with methods evolving from theoretically grounded models to deep learning approaches and their hybrids. Though promising, existing methods rarely address pansharpening through the lens of underlying physical imaging processes. In this work, we revisit the spectral imaging mechanism and propose a novel physics‐informed neural operator framework for pansharpening, termed PINO, which faithfully models the end‐to‐end electro‐optical sensor process. Specifically, PINO operates as: (1) First, a spatial-spectral encoder pair is introduced to aggregate multi-granularity high-resolution panchromatic (PAN) and low-resolution multispectral (LRMS) features. (2) Subsequently, an iterative neural integral process utilizes these fused spatial-spectral characteristics to learn a continuous radiance field $L_i(x, y, \lambda)$ over spatial coordinates and wavelength, effectively emulating band-wise spectral integration. (3) Finally, the learned radiance field is modulated by the sensor’s spectral responsivity $R_b(\lambda)$ to produce physically consistent spatial–spectral fusion products. This physics-grounded fusion paradigm offers a principled solution for reconstructing high-resolution multispectral and hyperspectral images in accordance with sensor imaging physics, effectively harnessing the unique advantages of spectral data to better uncover real-world characteristics. Experiments on multiple benchmark datasets show that our method surpasses state-of-the-art fusion algorithms, achieving reduced spectral aberrations and finer spatial textures. Furthermore, extension to hyperspectral (HS) data demonstrates its generalizability and universality. The code will be available upon potential acceptance.

Language models deployed in real-world systems often require post-hoc updates to incorporate new or corrected knowledge. However, editing such models efficiently and reliably—without retraining or forgetting previous information—remains a major challenge. Existing methods for lifelong model editing either compromise generalization, interfere with past edits, or fail to scale to long editing sequences. We propose MEMOIR, a novel scalable framework that injects knowledge through a residual memory, i.e., a dedicated parameter module, while preserving the core capabilities of the pre-trained model. By sparsifying input activations through data-dependent masks, MEMOIR confines each edit to a distinct subset of the memory parameters, minimizing interference among edits. At inference, it identifies relevant edits by comparing the sparse activation patterns of new queries to those stored during editing. This enables generalization to rephrased queries by activating only the relevant knowledge while suppressing unnecessary memory activation for unrelated prompts. Experiments on question answering, hallucination correction, and out-of-distribution generalization benchmarks across LLaMA-3 and Mistral demonstrate that MEMOIR achieves state-of-the-art performance across reliability, generalization, and locality metrics, scaling to thousands of sequential edits with minimal forgetting.

Graph Neural Networks (GNNs) have achieved impressive results across diverse network modeling tasks, but accurately estimating uncertainty on graphs remains difficult—especially under distributional shifts. Unlike traditional uncertainty estimation, graph-based uncertainty must account for randomness arising from both the graph’s structure and its label distribution, which adds complexity. In this paper, making an analogy between the evolution of a stochastic partial differential equation (SPDE) driven by Mat\'ern Gaussian Process and message passing using GNN layers, we present a principled way to design a novel message passing scheme that incorporates spatial-temporal noises motivated by the Gaussian Process approach to SPDE. Our method simultaneously captures uncertainty across space and time and allows explicit control over the covariance kernel’s smoothness, thereby enhancing uncertainty estimates on graphs with both low and high label informativeness. Our extensive experiments on Out-of-Distribution (OOD) detection on graph datasets with varying label informativeness demonstrate the soundness and superiority of our model to existing approaches.

Physics-informed neural networks (PINNs) have emerged as a promising framework to develop complex scientific surrogate models, yet their scalability and accuracy often degrade in non-canonical geometries, such as non-rectangular domains or three-dimensional (3D) domains with high aspect ratios. These limitations hinder the broader adoption of vanilla PINNs in real-world, practical systems. In this work, we introduce a multi-domain PINN (MDPINN) framework designed to address the scalability and generalization challenges inherent in 3D non-rectangular domains governed by nonlinear fluid dynamics. The target domain consists of intersecting 3D fluid channels with a high aspect ratio, inducing complex flow features such as deflections, mixing, and recirculations. Our approach is grounded in two key innovations: 1) domain decomposition, which partitions the channel volumes into multiple cubic-like subdomains, each modeled by an individual PINN, 2) enforcement of global dynamics (MDPINN-GD), which ensures that the total mass flow rate entering the domain equals that exiting. These innovations reduce the complexity of the problem imposed on individual PINNs and guide effective network optimization toward physically consistent solutions throughout the domain. We demonstrate that our method achieves: 1) 74.8\% accuracy improvement over a single-network PINN, and 2) 52.9\% accuracy improvement over MDPINN that do not enforce global mass conservation. Furthermore, the MDPINN-GD framework exhibits accurate prediction even in highly complex regions-such as the channel intersecting zone and the outlet zone characterized by intense flow mixing and large velocity gradients-achieving maximum normalized mean absolute errors below 14.9\% for velocity predictions compared to simulation results. This work establishes a path towards scalable, physically grounded surrogate modeling approach that is extensible to multiphysics and high-dimensional scientific problems.

How to integrate and verify spatial intelligence in foundation models remains an open challenge. Current practice often proxies Visual-Spatial Intelligence (VSI) with purely textual prompts and VQA-style scoring, which obscures geometry, invites linguistic shortcuts, and weakens attribution to genuinely spatial skills. We introduce Spatial Intelligence Grid (SIG): a structured, grid-based schema that explicitly encodes object layouts, inter-object relations, and physically grounded priors. As a complementary channel to text, SIG provides a faithful, compositional representation of scene structure for foundation-model reasoning. Building on SIG, we derive SIG-informed evaluation metrics that quantify a model’s intrinsic VSI, which separates spatial capability from language priors. In few-shot in-context learning with state-of-the-art multimodal LLMs (e.g. GPT- and Gemini-family models), SIG yields consistently larger, more stable, and more comprehensive gains across all VSI metrics compared to VQA-only representations, indicating its promise as a data-labeling and training schema for learning VSI. We also release SIGBench, a benchmark of 1.4K driving frames annotated with ground-truth SIG labels and human gaze traces, supporting both grid-based machine VSI tasks and attention-driven, human-like VSI tasks in autonomous-driving scenarios.

Inverse design aims to design the input variables of a physical system to optimize a specified objective function, typically formulated as a search or optimization problem. However, in 3D domains, the design space grows exponentially, rendering exhaustive grid-based searches infeasible. Recent advances in deep learning have accelerated inverse design by providing powerful generative priors and differentiable surrogate models. Nevertheless, current methods tend to approximate the 3D design space using 2D projections or fine-tune existing 3D shapes. These approaches sacrifice volumetric detail and constrain design exploration, preventing true 3D design from scratch. In this paper, we propose a 3D Inverse Design (3DID) framework that directly navigates the 3D design space by coupling a continuous latent representation with a physics-aware optimization strategy. We first learn a unified physics–geometry embedding that compactly captures shape and physical field data in a continuous latent space. Then, we introduce a two-stage strategy to perform physics-aware optimization. In the first stage, a gradient-guided diffusion sampler explores the global latent manifold. In the second stage, an objectivedriven, topology-preserving refinement further sculpts each candidate toward the target objective. This enables 3DID to generate high-fidelity 3D geometries, outperforming existing methods in both solution quality and design versatility.

Effectively leveraging prior knowledge of a system’s physics is crucial for applications of machine learning to scientific domains. Previous approaches mostly focused on incorporating physical insights at the architectural level. In this paper, we propose a framework to leverage physical information directly into the loss function for prediction and generative modeling tasks on systems like molecules and spins. We derive energy loss functions assuming that each data sample is in thermal equilibrium with respect to an approximate energy landscape. By using the reverse KL divergence with a Boltzmann distribution around the data, we obtain the loss as an energy difference between the data and the model predictions. This perspective also recasts traditional objectives like MSE as energy-based, but with a physically meaningless energy. In contrast, our formulation yields physically grounded loss functions with gradients that better align with valid configurations, while being architecture-agnostic and computationally efficient. The energy loss functions also inherently respect physical symmetries. We demonstrate our approach on molecular generation and spin ground-state prediction and report significant improvements over baselines.

Understanding architectural differences in language models is challenging, especially at academic-scale pretraining (e.g., 1.3B parameters, 100B tokens), where results are often dominated by noise and randomness. To overcome this, we introduce controlled synthetic pretraining tasks that isolate and evaluate core model capabilities. Within this framework, we discover \emph{Canon layers}: lightweight architectural components—named after the musical term ``canon''—that promote horizontal information flow across neighboring tokens. Canon layers compute weighted sums of nearby token representations and integrate seamlessly into Transformers, linear attention, state-space models, or any sequence architecture. We present 12 key results. This includes how Canon layers enhance reasoning depth (e.g., by $2\times$), reasoning breadth, knowledge manipulation, etc. They lift weak architectures like NoPE to match RoPE, and linear attention to rival SOTA linear models like Mamba2/GDN—validated both through synthetic tasks and real-world academic-scale pretraining. This synthetic playground offers an \emph{economical, principled path} to isolate core model capabilities often obscured at academic scales. Equipped with infinite high-quality data, it may even \emph{predict} how future architectures will behave as training pipelines improve—e.g., through better data curation or RL-based post-training—unlocking deeper reasoning and hierarchical inference.

Variational Quantum Algorithms (VQAs) are emerging as leading strategies with the potential to unlock practical applications and deliver significant advantages in the investigation of many-body quantum systems and quantum chemistry. A key challenge hindering the application of VQAs to large-scale problems is rooted in the no-cloning theorem in quantum mechanics, precluding standard backpropagation and leading to prohibitive quantum resource expenditure such as measurement cost. To address this challenge, we reformulate the training dynamics of VQAs as a non-linear partial differential equation and propose a novel protocol that leverages physics-informed neural networks (PINNs) to model this dynamical system efficiently. Given a small amount of training trajectory data collected from quantum devices, our protocol predicts the parameter updates of VQAs over multiple iterations on the classical side, dramatically reducing quantum resource costs. Through systematic numerical experiments, we demonstrate that our method achieves up to a 30x speedup compared to conventional methods and reduces quantum resource costs by as much as 90\% for tasks involving up to 40 qubits, including ground state preparation of different quantum systems, while maintaining competitive accuracy. Our approach complements existing techniques aimed at improving the efficiency of VQAs and further strengthens their potential for practical applications.

Physics-informed neural networks (PINNs) have shown significant promise in computational science and engineering, yet they often face optimization challenges and limited accuracy. In this work, we identify directional gradient conflicts during PINN training as a critical bottleneck. We introduce a novel gradient alignment score to systematically diagnose this issue through both theoretical analysis and empirical experiments. Building on these insights, we show that (quasi) second-order optimization methods inherently mitigate gradient conflicts, thereby consistently outperforming the widely used Adam optimizer. Among them, we highlight the effectiveness of SOAP \cite{vyas2024soap} by establishing its connection to Newton’s method. Empirically, SOAP achieves state-of-the-art results on 10 challenging PDE benchmarks, including the first successful application of PINNs to turbulent flows at Reynolds numbers up to 10,000. It yields 2–10x accuracy improvements over existing methods while maintaining computational scalability, advancing the frontier of neural PDE solvers for real-world, multi-scale physical systems. All code and datasets used in this work are publicly available at: \url{https://github.com/PredictiveIntelligenceLab/jaxpi/tree/pirate}. \end{abstract}