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.

Cross-LLM Consistency in Inference: Evidence from Shared Interactions

Large language models (LLMs) differ in architecture, training data, and optimization procedures, yet they may still develop similar internal inference patterns. In this paper, we examine this hypothesis using interaction-based explanations. We find that LLMs often share interaction patterns when predicting the same target token from the same prompt. This consistency is more pronounced among advanced LLMs. Shared interactions also tend to be lower-order and show weaker positive-negative cancellation than non-shared interactions. These results suggest that advanced LLMs may be implicitly optimized toward common inference patterns, even though the mechanisms that give rise to such cross-model consistency remain open.

A Geometry-Aware Triplane Field Network for Vehicle Aerodynamic Prediction

High-fidelity computational fluid dynamics (CFD) is crucial to vehicle aerodynamic analysis, but its cost still constrains early-stage design exploration. Machine-learning-based surface-field prediction offers a faster alternative if the model can efficiently capture both global flow context and local geometric detail. This work proposes a machine-learning-based method, named the geometry-aware triplane field network (GTF-Net), for vehicle aerodynamic pressure and wall shear stress prediction. GTF-Net constructs triplane features directly from sampled surface points through a shared multilayer perceptron (MLP) and smooth bilinear rasterization. The planes are then processed by a dual-stream backbone that combines adaptive Fourier neural operator (AFNO) spectral mixing with convolutional neural network (CNN) refinement, so long-range aerodynamic coupling and local geometry-induced variations are modeled in the same representation. At query stage, sampled triplane features are combined with vehicle-aligned directional coordinates, normal-projection features, and a voxel-based curvature proxy. GTF-Net is compared with Transolver, geometry-informed neural operator (GINO), and TripNet, a triplane-based surrogate model. GTF-Net improves the relative L2 error from the strongest baseline value of 0.157 to 0.145 for pressure prediction and from 0.237 to 0.226 for wall shear stress prediction. Ablation results show that AFNO mixing, local CNN refinement, and query-side geometric encoding each contribute to accuracy, supporting the proposed mechanism of combining structured triplane representation with explicit aerodynamic geometry cues.

CityTrajBench: A Unified Benchmark for City-Scale Vehicle Trajectory Generation

Urban trajectory generation is a fundamental task for transportation simulation, urban planning, and mobility analytics. However, systematic comparison across trajectory generation methods remains difficult because existing studies often rely on different datasets, preprocessing pipelines, trajectory representations, and evaluation metrics. This fragmentation makes it unclear whether reported performance differences arise from the generation mechanism itself or from inconsistent experimental protocols. To address this issue, we present CityTrajBench, a unified benchmark framework and protocol for city-scale vehicle trajectory generation. CityTrajBench standardizes data ingestion, trajectory normalization, feature construction, model adaptation, map-aware post-processing, model selection, and multi-level evaluation under a common setting. It supports heterogeneous generators, including statistical baselines, VAE-based, GAN-based, diffusion-based, and flow-matching-based models, and evaluates them on three real-world urban trajectory datasets. The benchmark measures global spatial realism, trip-level distribution fidelity, trajectory-level geometric similarity, conditional mobility consistency, and efficiency. Experiments reveal clear trade-offs across model families: DiffTraj is strongest on trajectory-level geometric fidelity, DiffRNTraj is competitive on structure-sensitive global realism, and TrajFlow provides a strong balance across realism, quality, conditional consistency, and efficiency. Meanwhile, a simple Markov baseline remains competitive on coarse-grained trip and local-movement statistics. These findings show that urban trajectory generation quality is inherently multi-objective, that no single model dominates all criteria equally, and that CityTrajBench provides a reproducible benchmark protocol and testbed for future research on urban mobility generation.

Graph-based data-driven discovery of interpretable laws governing corona-induced noise and radio interference for high-voltage transmission lines

The global shift towards renewable energy necessitates the development of ultrahigh-voltage (UHV) AC transmission to bridge the gap between remote energy sources and urban demand. While UHV grids offer superior capacity and efficiency, their implementation is often hindered by corona-induced audible noise (AN) and radio interference (RI). Since these emissions must meet strict environmental compliance standards, accurate prediction is vital for the large-scale deployment of UHV infrastructure. Existing engineering practices often rely on empirical laws, in which fixed log-linear structures limit accuracy and extrapolation. Herein, we present a monotonicity-constrained graph symbolic discovery framework, Mono-GraphMD, which uncovers compact, interpretable laws for corona-induced AN and RI. The framework provides mechanistic insight into how nonlinear interactions among the surface gradient, bundle number and diameter govern high-field emissions and enables accurate predictions for both corona-cage data and multicountry real UHV lines with up to 16-bundle conductors. Unlike black-box models, the discovered closed-form laws are highly portable and interpretable, allowing for rapid predictions when applied to various scenarios, thereby facilitating the engineering design process.

Inevitable Encounters: Backdoor Attacks Involving Lossy Compression

Real-world backdoor attacks often require poisoned datasets to be stored and transmitted before being used to compromise deep learning systems. However, in the era of big data, the inevitable use of lossy compression poses a fundamental challenge to invisible backdoor attacks. We find that triggers embedded in RGB images often become ineffective after the images are lossily compressed into binary bitstreams (e.g., JPEG files) for storage and transmission. As a result, the poisoned data lose its malicious effect after compression, causing backdoor injection to fail. In this paper, we highlight the necessity of explicitly accounting for the lossy compression process in backdoor attacks. This requires attackers to ensure that the transmitted binary bitstreams preserve malicious trigger information, so that effective triggers can be recovered in the decompressed data. Building on the region-of-interest (ROI) coding mechanism in image compression, we propose two poisoning strategies tailored to inevitable lossy compression. First, we introduce Universal Attack Activation, a universal method that uses sample-specific ROI masks to reactivate trigger information in binary bitstreams for learned image compression (LIC). Second, we present Compression-Adapted Attack, a new attack strategy that employs customized ROI masks to encode trigger information into binary bitstreams and is applicable to both traditional codecs and LIC. Extensive experiments demonstrate the effectiveness of both strategies.

GENSR: Symbolic Regression Based in Equation Generative Space

Symbolic Regression (SR) tries to reveal the hidden equations behind observed data. However, most methods search within a discrete equation space, where the structural modifications of equations rarely align with their numerical behavior, leaving fitting error feedback too noisy to guide exploration. To address this challenge, we propose GenSR, a generative latent space-based SR framework following the `map construction -> coarse localization -> fine search'' paradigm. Specifically, GenSR first pretrains a dual-branch Conditional Variational Autoencoder (CVAE) to reparameterize symbolic equations into a generative latent space with symbolic continuity and local numerical smoothness. This space can be regarded as a well-structured `map'' of the equation space, providing directional signals for search. At inference, the CVAE coarsely localizes the input data to promising regions in the latent space. Then, a modified CMA-ES refines the candidate region, leveraging smooth latent gradients. From a Bayesian perspective, GenSR reframes the SR task as maximizing the conditional distribution $p(\mathrm{Equ.} \mid \mathrm{Num.})$, with CVAE training achieving this objective through the Evidence Lower Bound (ELBO). This new perspective provides a theoretical guarantee for the effectiveness of GenSR. Extensive experiments show that GenSR jointly optimizes predictive accuracy, expression simplicity, and computational efficiency, while remaining robust under noise.

Dynamic Training-Free Fusion of Subject and Style LoRAs

Recent studies have explored the combination of multiple LoRAs to simultaneously generate user-specified subjects and styles. However, most existing approaches fuse LoRA weights using static statistical heuristics that deviate from LoRA's original purpose of learning adaptive feature adjustments and ignore the randomness of sampled inputs. To address this, we propose a dynamic training-free fusion framework that operates throughout the generation process. During the forward pass, at each LoRA-applied layer, we dynamically compute the KL divergence between the base model's original features and those produced by subject and style LoRAs, respectively, and adaptively select the most appropriate weights for fusion. In the reverse denoising stage, we further refine the generation trajectory by dynamically applying gradient-based corrections derived from objective metrics such as CLIP and DINO scores, providing continuous semantic and stylistic guidance. By integrating these two complementary mechanisms-feature-level selection and metric-guided latent adjustment-across the entire diffusion timeline, our method dynamically achieves coherent subject-style synthesis without any retraining. Extensive experiments across diverse subject-style combinations demonstrate that our approach consistently outperforms state-of-the-art LoRA fusion methods both qualitatively and quantitatively.

Learning Domain Knowledge in Multimodal Large Language Models through Reinforcement Fine-Tuning

Multimodal large language models (MLLMs) have shown remarkable capabilities in multimodal perception and understanding tasks. However, their effectiveness in specialized domains, such as remote sensing and medical imaging, remains limited. A natural approach to domain adaptation is to inject domain knowledge through textual instructions, prompts, or auxiliary captions. Surprisingly, we find that such input-level domain knowledge injection yields little to no improvement on scientific multimodal tasks, even when the domain knowledge is explicitly provided. This observation suggests that current MLLMs fail to internalize domain-specific priors through language alone, and that domain knowledge must be integrated at the optimization level. Motivated by this insight, we propose a reinforcement fine-tuning framework that incorporates domain knowledge directly into the learning objective. Instead of treating domain knowledge as descriptive information, we encode it as domain-informed constraints and reward signals, shaping the model's behavior in the output space. Extensive experiments across multiple datasets in remote sensing and medical domains consistently demonstrate good performance gains, achieving state-of-the-art results on multimodal domain tasks. Our results highlight the necessity of optimization-level domain knowledge integration and reveal a fundamental limitation of textual domain conditioning in current MLLMs.

Beyond empirical models: Discovering new constitutive laws in solids with graph-based equation discovery

Constitutive models are fundamental to solid mechanics and materials science, underpinning the quantitative description and prediction of material responses under diverse loading conditions. Traditional phenomenological models, which are derived through empirical fitting, often lack generalizability and rely heavily on expert intuition and predefined functional forms. In this work, we propose a graph-based equation discovery framework for the automated discovery of constitutive laws directly from multisource experimental data. This framework expresses equations as directed graphs, where nodes represent operators and variables, edges denote computational relations, and edge features encode parametric dependencies. This enables the generation and optimization of free-form symbolic expressions with undetermined material-specific parameters. Through the proposed framework, we have discovered new constitutive models for strain-rate effects in alloy steel materials and the deformation behavior of lithium metal. Compared with conventional empirical models, these new models exhibit compact analytical structures and achieve higher accuracy. The proposed graph-based equation discovery framework provides a generalizable and interpretable approach for data-driven scientific modelling, particularly in contexts where traditional empirical formulations are inadequate for representing complex physical phenomena.