Estimating probability density and its score from samples remains a core problem in generative modeling, Bayesian inference, and kinetic theory. Existing methods are bifurcated: classical kernel density estimators (KDE) generalize across distributions but suffer from the curse of dimensionality, while modern neural score models achieve high precision but require retraining for every target distribution. We introduce DiScoFormer (Density and Score Transformer), a ``train-once, infer-anywhere" equivariant Transformer that maps i.i.d. samples to both density values and score vectors, generalizing across distributions and sample sizes. Analytically, we prove that self-attention can recover normalized KDE, establishing it as a functional generalization of kernel methods; empirically, individual attention heads learn multi-scale, kernel-like behaviors. The model converges faster and achieves higher precision than KDE for density estimation, and provides a high-fidelity plug-in score oracle for score-debiased KDE, Fisher information computation, and Fokker-Planck-type PDEs.

High-fidelity measurements of continuum physical fields are essential for scientific discovery and engineering design but remain challenging under sparse and constrained sensing. Conventional reconstruction methods typically rely on fixed sensor layouts, which cannot adapt to evolving physical states. We propose LASER, a unified, closed-loop framework that formulates active sensing as a Partially Observable Markov Decision Process (POMDP). At its core, LASER employs a continuum field latent world model that captures the underlying physical dynamics and provides intrinsic reward feedback. This enables a reinforcement learning policy to simulate ''what-if'' sensing scenarios within a latent imagination space. By conditioning sensor movements on predicted latent states, LASER navigates toward potentially high-information regions beyond current observations. Our experiments demonstrate that LASER consistently outperforms static and offline-optimized strategies, achieving high-fidelity reconstruction under sparsity across diverse continuum fields.

Multimodal learning aims to integrate multi-sensor data to exploit their complementary information, embracing a more comprehensive real-world perception and understanding. However, heterogeneous discrepancies across modalities consistently trigger imbalanced multimodal optimization, restricting the joint learning performance. Although existing methods mitigate this issue through optimization modulation and conflict alleviation, they still suffer from entangled optimization and uniform learning pace in conventional monolithic frameworks, limiting the effectiveness of multimodal learning. To address this issue, we propose a novel Multimodal Nested Learning Framework (MoNet), which reformulates the monolithic framework into nested sub-processes, decoupling and coordinating multimodal learning. To achieve this, we present a Decoupled Multimodal Stable Memory block (DMSM) as the outermost nested level, which decouples multimodal learning into independent optimization streams for semantic exploitation across modalities. Additionally, we develop an Adaptive Multimodal Coordinated Fusion block (AMCF), which constitutes the inner nested level. It attempts to coordinate multimodal information integration across multi-timescale nested memories, balancing multimodal fusion. Extensive experimental results on eight datasets across three tasks demonstrate the superiority of MoNet. Code is available at https://github.com/Yangl1nFeng/MoNet.

High-dimensional datasets often concentrate near low-dimensional structures, but estimating their geometry from samples typically relies on graphs and kernels that scale poorly with dataset size and dimension. We propose **Riemannian metric matching**: a denoising probabilistic framework for learning the Riemannian geometry of data using neural networks. Specifically, we learn the *carré du champ* operator, which, using diffusion geometry, gives us access to the Riemannian geometry toolkit for downstream machine learning and statistical tasks. Our key observation is that the carré du champ operator can be formulated as a conditional expectation over random perturbations of the data, which can be exploited for sample-wise training and constant cost, amortized inference without explicit kernel construction. Empirically, metric matching rivals or improves the accuracy of $k$-NN-based diffusion geometry estimators, while enabling amortized inference that is up to $400\times$ faster, and supports graph-free geometric analysis on high-dimensional images where nearest neighbors break down.