Governing equation discovery of a complex system from snapshots

Complex systems in physics, chemistry, and biology that evolve over time with inherent randomness are typically described by stochastic differential equations (SDEs). A fundamental challenge in science and engineering is to determine the governing equations of a complex system from snapshot data. Traditional equation discovery methods often rely on stringent assumptions, such as the availability of the trajectory information or time-series data, and the presumption that the underlying system is deterministic. In this work, we introduce a data-driven, simulation-free framework, called Sparse Identification of Differential Equations from Snapshots (SpIDES), that discovers the governing equations of a complex system from snapshots by utilizing the advanced machine learning techniques to perform three essential steps: probability flow reconstruction, probability density estimation, and Bayesian sparse identification. We validate the effectiveness and robustness of SpIDES by successfully identifying the governing equation of an over-damped Langevin system confined within two potential wells. By extracting interpretable drift and diffusion terms from the SDEs, our framework provides deeper insights into system dynamics, enhances predictive accuracy, and facilitates more effective strategies for managing and simulating stochastic systems.

Learning Hamiltonian neural Koopman operator and simultaneously sustaining and discovering conservation law

Accurately finding and predicting dynamics based on the observational data with noise perturbations is of paramount significance but still a major challenge presently. Here, for the Hamiltonian mechanics, we propose the Hamiltonian Neural Koopman Operator (HNKO), integrating the knowledge of mathematical physics in learning the Koopman operator, and making it automatically sustain and even discover the conservation laws. We demonstrate the outperformance of the HNKO and its extension using a number of representative physical systems even with hundreds or thousands of freedoms. Our results suggest that feeding the prior knowledge of the underlying system and the mathematical theory appropriately to the learning framework can reinforce the capability of machine learning in solving physical problems.

From Fourier to Neural ODEs: Flow Matching for Modeling Complex Systems

Modeling complex systems using standard neural ordinary differential equations (NODEs) often faces some essential challenges, including high computational costs and susceptibility to local optima. To address these challenges, we propose a simulation-free framework, called Fourier NODEs (FNODEs), that effectively trains NODEs by directly matching the target vector field based on Fourier analysis. Specifically, we employ the Fourier analysis to estimate temporal and potential high-order spatial gradients from noisy observational data. We then incorporate the estimated spatial gradients as additional inputs to a neural network. Furthermore, we utilize the estimated temporal gradient as the optimization objective for the output of the neural network. Later, the trained neural network generates more data points through an ODE solver without participating in the computational graph, facilitating more accurate estimations of gradients based on Fourier analysis. These two steps form a positive feedback loop, enabling accurate dynamics modeling in our framework. Consequently, our approach outperforms state-of-the-art methods in terms of training time, dynamics prediction, and robustness. Finally, we demonstrate the superior performance of our framework using a number of representative complex systems.

Rethinking of Feature Interaction for Multi-task Learning on Dense Prediction

Existing works generally adopt the encoder-decoder structure for Multi-task Dense Prediction, where the encoder extracts the task-generic features, and multiple decoders generate task-specific features for predictions. We observe that low-level representations with rich details and high-level representations with abundant task information are not both involved in the multi-task interaction process. Additionally, low-quality and low-efficiency issues also exist in current multi-task learning architectures. In this work, we propose to learn a comprehensive intermediate feature globally from both task-generic and task-specific features, we reveal an important fact that this intermediate feature, namely the bridge feature, is a good solution to the above issues. Based on this, we propose a novel Bridge-Feature-Centirc Interaction (BRFI) method. A Bridge Feature Extractor (BFE) is designed for the generation of strong bridge features and Task Pattern Propagation (TPP) is applied to ensure high-quality task interaction participants. Then a Task-Feature Refiner (TFR) is developed to refine final task predictions with the well-learned knowledge from the bridge features. Extensive experiments are conducted on NYUD-v2 and PASCAL Context benchmarks, and the superior performance shows the proposed architecture is effective and powerful in promoting different dense prediction tasks simultaneously.

Rethinking Cross-Domain Pedestrian Detection: A Background-Focused Distribution Alignment Framework for Instance-Free One-Stage Detectors

Cross-domain pedestrian detection aims to generalize pedestrian detectors from one label-rich domain to another label-scarce domain, which is crucial for various real-world applications. Most recent works focus on domain alignment to train domain-adaptive detectors either at the instance level or image level. From a practical point of view, one-stage detectors are faster. Therefore, we concentrate on designing a cross-domain algorithm for rapid one-stage detectors that lacks instance-level proposals and can only perform image-level feature alignment. However, pure image-level feature alignment causes the foreground-background misalignment issue to arise, i.e., the foreground features in the source domain image are falsely aligned with background features in the target domain image. To address this issue, we systematically analyze the importance of foreground and background in image-level cross-domain alignment, and learn that background plays a more critical role in image-level cross-domain alignment. Therefore, we focus on cross-domain background feature alignment while minimizing the influence of foreground features on the cross-domain alignment stage. This paper proposes a novel framework, namely, background-focused distribution alignment (BFDA), to train domain adaptive onestage pedestrian detectors. Specifically, BFDA first decouples the background features from the whole image feature maps and then aligns them via a novel long-short-range discriminator.