DPMs, those inherent factors can be automatically discovered, explicitly represented, and clearly injected into the diffusion process via the sub-gradient fields. To tackle this task, we devise an unsupervised approach named DisDiff, achieving disentangled representation learning in the framework of DPMs.

The contributions of this paper are as follows (Figure 1): 1.

We prove that if these charges flow upward along electric field lines, their initial distribution in the z = 0 plane transforms into a distribution on the hemisphere of radiusr that becomes uniform in the r limit. To learn the bijective transformation, we estimate the normalized field in the augmented space.

The aim of this work is to learn models of population dynamics of physical systems that feature stochastic and mean-field effects and that depend on physics parameters. The learned models can act as surrogates of classical numerical models to efficiently predict the system behavior over the physics parameters. Building on the Benamou-Brenier formula from optimal transport and action matching, we use a variational problem to infer parameter-and time-dependent gradient fields that represent approximations of the population dynamics. The inferred gradient fields can then be used to rapidly generate sample trajectories that mimic the dynamics of the physical system on a population level over varying physics parameters. We show that combining Monte Carlo sampling with higher-order quadrature rules is critical for accurately estimating the training objective from sample data and for stabilizing the training process. We demonstrate on Vlasov-Poisson instabilities as well as on high-dimensional particle and chaotic systems that our approach accurately predicts population dynamics over a wide range of parameters and outperforms state-of-the-art diffusion-based and flow-based modeling that simply condition on time and physics parameters.

The main goal of generative modeling is to use finitely many samples from a distribution to construct a sampling scheme capable of generating new samples from the same distribution. Among the families of existing generative models, flow matching (FM) [23, 24] is notable for its flexibility and simplicity. Given a target probability distribution, FM utilizes a parametric model (e.g., neural network) to learn the velocity vector field that defines a deterministic, continuous transformation (a normalizing flow) and transports a source probability distribution (e.g., standard Gaussian) to the target distribution. While the population formulation of FM often exhibits appealing structure--sometimes even admitting gradient-field velocities--practical models are trained on finite datasets and therefore optimize empirical objectives. This empirical setting substantially alters the geometry of the learned velocity field and the energetic properties of the resulting sampler. These notes aim to clarify how empirical FM behaves, how it differs from its population counterpart, and what implicit biases arise in the learned sampling dynamics. From now on, we assume that all the probability distributions/measures (except the empirical distribution) of the random variables considered are absolutely continuous (i.e., they have densities with respect to the Lebesgue measure), in which case we shall abuse the notation and use the same symbol to denote both the distribution and the density. To maintain the flow of the main text, we defer discussion of related work and all proofs of the theoretical results to the appendix.

In critical web applications such as e-commerce and recommendation systems, multimodal graphs integrating rich visual and textual attributes are increasingly central, yet their large scale introduces substantial computational burdens for training Graph Neural Networks (GNNs). While Graph Condensation (GC) offers a promising solution by synthesizing smaller datasets, existing methods falter in the multimodal setting. We identify a dual challenge causing this failure: (1) conflicting gradients arising from semantic misalignments between modalities, and (2) the GNN's message-passing architecture pathologically amplifying this gradient noise across the graph structure. To address this, we propose Structurally-Regularized Gradient Matching (SR-GM), a novel condensation framework tailored for multimodal graphs. SR-GM introduces two synergistic components: first, a gradient decoupling mechanism that resolves inter-modality conflicts at their source via orthogonal projection; and second, a structural damping regularizer that acts directly on the gradient field. By leveraging the graph's Dirichlet energy, this regularizer transforms the topology from a noise amplifier into a stabilizing force during optimization. Extensive experiments demonstrate that SR-GM significantly improves accuracy and accelerates convergence compared to baseline methods. Ablation studies confirm that addressing both gradient conflict and structural amplification in tandem is essential for achieving superior performance. Moreover, the condensed multimodal graphs exhibit strong cross-architecture generalization and promise to accelerate applications like Neural Architecture Search. This research provides a scalable methodology for multimodal graph-based learning in resource-constrained environments.

The contributions of this paper are as follows (Figure 1): 1.