Data-driven discovery of governing differential equations across physical systems

Differential equations play a critical role in scientific discovery because they provide a mathematical framework to describe the behaviour of physical phenomena. As a promising alternative to traditional first principles, data-driven differential equation discovery has attracted increasing attention for its ability to infer governing laws directly from experimental or simulated data, especially when the underlying physics is unclear. However, the field has expanded rapidly along diverse methodological directions, particularly with the emergence of AI-based approaches, and still lacks a clear organizing perspective. In this Review, we propose a problem-oriented perspective on data-driven differential equation discovery. We first introduce a two-dimensional phase diagram of equation discoverability, where discovery problems are organized according to structural complexity and coefficient complexity. This phase diagram shows how the field has moved from the discovery of sparse equations with simple coefficients toward more complex governing laws with richer structures and more flexible parameterizations. It also clarifies why different methodological families succeed or fail in different problem settings. We then present the representation-evaluation-optimization (REO) framework as a fundamental abstraction of the discovery process. By identifying the core problems of equation discovery that persist across algorithmic variations, REO shifts the discussion from individual algorithms to the fundamental principles that determine discoverability. We connect these perspectives to applications across physics and adjacent sciences, and argue that the next challenge is not merely recovering equations, but using them to revise existing theories, distil mechanisms and form new scientific concepts.

Tractogram foundation model

Diffusion MRI (dMRI) tractography is the only noninvasive approach for mapping white-matter pathways in the living human brain. It represents each brain as a tractogram: a large, unordered set of three-dimensional streamlines that includes information about both local streamline geometry and whole-brain anatomical organization. This structure makes tractograms a natural but challenging target for representation learning. Existing methods treat streamline classification and subject-level prediction as separate problems: streamline classifiers focus on geometric patterns, whereas subject-level prediction often depends on hand-crafted features. As a result, current methods do not learn reusable representations that connect streamline anatomy with whole-brain inter-subject variation. Here we introduce TractFM, a tractogram foundation model that learns reusable representations directly from whole-brain streamline sets. TractFM combines a local streamline encoder with a permutation-equivariant tractogram encoder, allowing all streamlines from a subject to be contextualized jointly in a single forward pass. Pretraining on dense anatomical tract parcellation, i.e., assigning anatomical labels to individual streamlines, yields two complementary representations: contextualized streamline-level embeddings for tract parcellation and compact subject-level descriptors for downstream prediction of subject phenotypes. Across three tractography algorithms and five dMRI datasets, TractFM transfers to both streamline-level and subject-level tasks. Its frozen representations achieve accurate tract parcellation and predict age and sex across independent datasets. These results show that whole-brain geometric context, learned once, can generalize across tractography pipelines, datasets, and prediction tasks.

3D Gaussian Map with Open-Set Semantic Grouping for Vision-Language Navigation

Vision-language navigation (VLN) requires an agent to traverse complex 3D environments based on natural language instructions, necessitating a thorough scene understanding. While existing works equip agents with various scene representations to enhance spatial awareness, they often neglect the complex 3D geometry and rich semantics in VLN scenarios, limiting the ability to generalize across diverse and unseen environments. To address these challenges, this work proposes a 3D Gaussian Map that represents the environment as a set of differentiable 3D Gaussians and accordingly develops a navigation strategy for VLN. Specifically, Egocentric Scene Map is constructed online by initializing 3D Gaussians from sparse pseudo-lidar point clouds, providing informative geometric priors for scene understanding. Each Gaussian primitive is further enriched through Open-Set Semantic Grouping operation, which groups 3D Gaussians based on their membership in object instances or stuff categories within the open world, resulting in a unified 3D Gaussian Map. Building on this map, Multi-Level Action Prediction strategy, which combines spatial-semantic cues at multiple granularities, is designed to assist agents in decision-making. Extensive experiments conducted on three public benchmarks (i.e., R2R, R4R, and REVERIE) validate the effectiveness of our method.

Uncertainty-Aware Gaussian Map for Vision-Language Navigation

Vision-Language Navigation (VLN) requires an agent to navigate 3D environments following natural language instructions. During navigation, existing agents commonly encounter perceptual uncertainty, such as insufficient evidence for reliable grounding or ambiguity in interpreting spatial cues, yet they typically ignore such information when predicting actions. In this work, we explicitly model three forms of perceptual uncertainty (i.e., geometric, semantic, and appearance uncertainty) and integrate them into the agent's observation space to enable informed decision-making. Concretely, our agent first constructs a Semantic Gaussian Map (SGM), composed of differentiable 3D Gaussian primitives initialized from panoramic observations, that encodes both the geometric structure and semantic content of the environment. On top of SGM, geometric uncertainty is estimated through variational perturbations of Gaussian position and scale to assess structural reliability; semantic uncertainty is captured by perturbing Gaussian semantic attributes to reveal ambiguous interpretations; and appearance uncertainty is characterized by Fisher Information, which measures the sensitivity of rendered observations to Gaussian-level variations. These uncertainties are incorporated into SGM, extending it into a unified 3D Value Map, which grounds them as affordances and constraints that support reliable navigation. Comprehensive evaluations across multiple VLN benchmarks show the effectiveness of our agent.

Clinically-Grounded Counterfactual Reasoning for Medical Video Diagnosis

Medical video diagnosis involves inferring clinical decisions from dynamic tissue responses throughout examination processes. Existing methods rely on an end-to-end learning paradigm that i) focuses on appearance rather than pathology, ii) lacks clinical priors, and iii) reasons solely from observations without counterfactual comparison. This work introduces MedVCR, a counterfactual reasoning framework that mimics clinical diagnostic thinking. MedVCR comprises three components: a Counterfactual Generator that synthesizes tissue evolution under specified pathological states via a diffusion-based manner; a Counterfactual Representation Learning module that encodes diagnostic knowledge through clinical rules (i.e., temporal consistency, pathological separability, and counterfactual alignment); and a Dual Diagnostic Prediction strategy that integrates video-level assessment with frame-level counterfactual analysis. MedVCR is evaluated under both fully supervised (e.g., colposcopy) and weakly supervised (e.g., colonoscopy) video diagnosis settings, yielding 2.6%-10.2% performance gains compared with leading baselines. Comprehensive ablation studies further validate the effectiveness of each component. The code will be released.

AxiomOcean: Forecasting the Three-Dimensional Structure of the Upper Ocean

Short-term ocean forecast skill depends strongly on the three-dimensional ocean structure of the upper ocean, which governs stratification, subsurface heat storage, and the response of the ocean to atmospheric forcing. However, AI ocean forecasting models often fail to preserve this vertical structure, resulting in over-smoothed subsurface features and weak physical consistency under strong forcing. Here, we present AxiomOcean, a global AI ocean forecasting model that explicitly represents vertical hierarchy and cross-layer dependence within the water column. By combining a fully three-dimensional encoder-backbone-decoder architecture with surface atmospheric forcing, AxiomOcean jointly predicts upper-ocean temperature, salinity, and three-dimensional currents at global 1/12° resolution down to 643 m depth. In 10-day forecasts, AxiomOcean outperforms an advanced AI comparison model across variables and lead times, reducing day-1 RMSE by approximately 20 to 35% while maintaining higher anomaly correlation. The gain is not achieved through excessive smoothing: AxiomOcean better preserves eddy kinetic energy, temperature and salinity variance. Its advantage also extends through the water column and remains evident across the equatorial Pacific, Kuroshio Extension, and Southern Ocean, yielding a more realistic reconstruction of upper-ocean heat content. These results show that explicitly preserving upper-ocean three-dimensional structure can improve both forecast accuracy and physical fidelity in AI ocean prediction.

Learning 3D Representations for Spatial Intelligence from Unposed Multi-View Images

Robust 3D representation learning forms the perceptual foundation of spatial intelligence, enabling downstream tasks in scene understanding and embodied AI. However, learning such representations directly from unposed multi-view images remains challenging. Recent self-supervised methods attempt to unify geometry, appearance, and semantics in a feed-forward manner, but they often suffer from weak geometry induction, limited appearance detail, and inconsistencies between geometry and semantics. We introduce UniSplat, a feed-forward framework designed to address these limitations through three complementary components. First, we propose a dual-masking strategy that strengthens geometry induction in the encoder. By masking both encoder and decoder tokens, and targeting decoder masks toward geometry-rich regions, the model is forced to infer structural information from incomplete visual cues, yielding geometry-aware representations even under unposed inputs. Second, we develop a coarse-to-fine Gaussian splatting strategy that reduces appearance-semantics inconsistencies by progressively refining the radiance field. Finally, to enforce geometric-semantic consistency, we introduce a pose-conditioned recalibration mechanism that interrelates the outputs of multiple heads by re-projecting predicted 3D point and semantic maps into the image plane using estimated camera parameters, and aligning them with corresponding RGB and semantic predictions to ensure cross-task consistency, thereby resolving geometry-semantic mismatches. Together, these components yield unified 3D representations that are robust to unposed, sparse-view inputs and generalize across diverse tasks, laying a perceptual foundation for spatial intelligence.

SinkTrack: Attention Sink based Context Anchoring for Large Language Models

Large language models (LLMs) suffer from hallucination and context forgetting. Prior studies suggest that attention drift is a primary cause of these problems, where LLMs' focus shifts towards newly generated tokens and away from the initial input context. To counteract this, we make use of a related, intrinsic characteristic of LLMs: attention sink -- the tendency to consistently allocate high attention to the very first token (i.e., <BOS>) of a sequence. Concretely, we propose an advanced context anchoring method, SinkTrack, which treats <BOS> as an information anchor and injects key contextual features (such as those derived from the input image or instruction) into its representation. As such, LLM remains anchored to the initial input context throughout the entire generation process. SinkTrack is training-free, plug-and-play, and introduces negligible inference overhead. Experiments demonstrate that SinkTrack mitigates hallucination and context forgetting across both textual (e.g., +21.6% on SQuAD2.0 with Llama3.1-8B-Instruct) and multi-modal (e.g., +22.8% on M3CoT with Qwen2.5-VL-7B-Instruct) tasks. Its consistent gains across different architectures and scales underscore the robustness and generalizability. We also analyze its underlying working mechanism from the perspective of information delivery. Our source code is available at https://github.com/67L1/SinkTrack.

PKINet-v2: Towards Powerful and Efficient Poly-Kernel Remote Sensing Object Detection

Object detection in remote sensing images (RSIs) is challenged by the coexistence of geometric and spatial complexity: targets may appear with diverse aspect ratios, while spanning a wide range of object sizes under varied contexts. Existing RSI backbones address the two challenges separately, either by adopting anisotropic strip kernels to model slender targets or by using isotropic large kernels to capture broader context. However, such isolated treatments lead to complementary drawbacks: the strip-only design can disrupt spatial coherence for regular-shaped objects and weaken tiny details, whereas isotropic large kernels often introduce severe background noise and geometric mismatch for slender structures. In this paper, we extend PKINet, and present a powerful and efficient backbone that jointly handles both challenges within a unified paradigm named Poly Kernel Inception Network v2 (PKINet-v2). PKINet-v2 synergizes anisotropic axial-strip convolutions with isotropic square kernels and builds a multi-scope receptive field, preserving fine-grained local textures while progressively aggregating long-range context across scales. To enable efficient deployment, we further introduce a Heterogeneous Kernel Re-parameterization (HKR) Strategy that fuses all heterogeneous branches into a single depth-wise convolution for inference, eliminating fragmented kernel launches without accuracy loss. Extensive experiments on four widely-used benchmarks, including DOTA-v1.0, DOTA-v1.5, HRSC2016, and DIOR-R, demonstrate that PKINet-v2 achieves state-of-the-art accuracy while delivering a $\textbf{3.9}\times$ FPS acceleration compared to PKINet-v1, surpassing previous remote sensing backbones in both effectiveness and efficiency.

Iris: Bringing Real-World Priors into Diffusion Model for Monocular Depth Estimation

In this paper, we propose \textbf{Iris}, a deterministic framework for Monocular Depth Estimation (MDE) that integrates real-world priors into the diffusion model. Conventional feed-forward methods rely on massive training data, yet still miss details. Previous diffusion-based methods leverage rich generative priors yet struggle with synthetic-to-real domain transfer. Iris, in contrast, preserves fine details, generalizes strongly from synthetic to real scenes, and remains efficient with limited training data. To this end, we introduce a two-stage Priors-to-Geometry Deterministic (PGD) schedule: the prior stage uses Spectral-Gated Distillation (SGD) to transfer low-frequency real priors while leaving high-frequency details unconstrained, and the geometry stage applies Spectral-Gated Consistency (SGC) to enforce high-frequency fidelity while refining with synthetic ground truth. The two stages share weights and are executed with a high-to-low timestep schedule. Extensive experimental results confirm that Iris achieves significant improvements in MDE performance with strong in-the-wild generalization.