Principal stratification provides a foundational framework for causal inference with intermediate outcomes by defining causal effects within subpopulations, yet existing work has largely focused on average effects across strata rather than treatment effect heterogeneity within strata. Such within-stratum heterogeneity informs individualized treatment decisions but the associated methods are sparse. We address this gap by studying the identification and estimation of the conditional principal causal effects under principal ignorability combined with an odds ratio sensitivity parameterization, which relaxes the monotonicity assumption. To efficiently learn these estimands, we propose a novel doubly cross-fit doubly robust machine learner that resolves the nested nuisance structure inherent to principal stratification. Leveraging sequential orthogonal learning with regularized least-squares sieves, we derive $\mathcal{L}^2$ and uniform limit theory, establish oracle efficiency, and construct uniform confidence bands for the proposed estimator. We use simulations to demonstrate the finite-sample performance of our estimator, and provide an empirical analysis of a randomized trial in acute lung injury, revealing informative patterns of treatment effect heterogeneity within the always-survivor subpopulation.

Many modeling tasks from disparate domains can be framed in the same way, computing spherical signals from geometric inputs, for example, computing the radar response of different objects or navigating through an environment. This paper introduces G2Sphere, a general method for mapping object geometries to spherical signals. G2Sphere operates entirely in Fourier space, encoding geometric structure into latent Fourier features using equivariant neural networks and outputting the Fourier coefficients of the continuous target signal, which can be evaluated at any resolution. By utilizing a hybrid GNN-spherical CNN architecture, our method achieves a much higher frequency output signal than comparable equivariant GNNs and avoids hand-engineered geometry features used previously by purely spherical methods. We perform experiments on various challenging domains, including radar response modeling, aerodynamic drag prediction, and policy learning for manipulation and navigation. We find that G2Sphere outperforms competitive baselines in terms of accuracy and inference time, and we demonstrate that equivariance and Fourier features lead to improved sample efficiency and generalization.

We introduce a semi-parametric framework for nonlinear system identification, which decouples discrepancy functions from physics-based components. Orthogonal Gaussian process regression balances sparse parameter selection (the white box) with discrepancy learning (the black box) to produce interpretable models from incomplete physics.

We propose a novel online Gaussian process (GP) model that is capable of capturing long-term memory in sequential data in an online learning setting. Our model, Online HiPPO Sparse Variational Gaussian Process (OHSVGP), leverages the HiPPO (High-order Polynomial Projection Operators) framework, which is popularized in the RNN domain due to its long-range memory modeling capabilities. We interpret the HiPPO time-varying orthogonal projections as inducing variables with timedependent orthogonal polynomial basis functions, which allows the SVGP inducing variables to memorize the process history. We show that the HiPPO framework fits naturally into the interdomain GP framework and demonstrate that the kernel matrices can also be updated online in a recurrence form based on the ODE evolution of HiPPO. We evaluate OHSVGP with online prediction for 1D time series, continual learning in discriminative GP model for data with multidimensional inputs, and deep generative modeling with sparse Gaussian process variational autoencoder, showing that it outperforms existing online GP methods in terms of predictive performance, long-term memory preservation, and computational efficiency.

We study a shallow variant of XNet, a neural architecture whose activation functions are derived from the Cauchy integral formula. While prior work focused on deep variants, we show that even a single-layer XNet exhibits near-exponential approximation rates--exceeding the polynomial bounds of MLPs and spline-based networks such as Kolmogorov-Arnold Networks (KANs). Empirically, XNet reduces approximation error by over 600 on discontinuous functions, achieves up to 20,000 lower residuals in physics-informed PDEs, and improves policy accuracy and sample efficiency in PPO-based reinforcement learning--while maintaining comparable or better computational efficiency than KAN baselines. These results demonstrate that expressive approximation can stem from principled activation design rather than depth alone, offering a compact, theoretically grounded alternative for function approximation, scientific computing, and control.

Vehicle trajectory prediction serves as a critical enabler for autonomous navigation and intelligent transportation systems. While existing approaches predominantly focus on pattern extraction and vehicle-environment interaction modeling, they exhibit a fundamental limitation in addressing trajectory heterogeneity originating from human driving styles. This oversight constrains prediction reliability in complex real-world scenarios. To bridge this gap, we propose the Driving-StyleAdaptive (DSA) framework, which establishes the first systematic integration of heterogeneous driving behaviors into trajectory prediction models. Specifically, our framework employs a set of basis functions tailored to each driving style to approximate the trajectory patterns. By dynamically combining and adaptively adjusting the degree of these basis functions, DSA not only enhances prediction accuracy but also provides explanations insights into the prediction process. Extensive experiments on public real-world datasets demonstrate that the DSA framework outperforms state-of-the-art methods.

Kolmogorov-Arnold Networks (KANs) have demonstrated remarkable expressive capacity and predictive power in symbolic learning. However, existing generalization errors of KANs primarily focus on approximation errors while neglecting estimation errors, leading to a suboptimal bias-variance trade-off and poor generalization performance. Meanwhile, the unclear generalization mechanism hinders the design of more effective KANs. As the authors of KANs highlighted, they "would like to explore ways to restrict KANs' hypothesis space so that they can achieve good performance." To address these challenges, we explore the generalization mechanism of KANs and design more effective KANs with lower model complexity and better generalization. We define Lipschitz complexity as the first structural measure for deep functions represented by KANs and derive novel generalization bounds based on Lipschitz complexity, establishing a theoretical foundation for understanding their generalization behavior. To reduce Lipschitz complexity and boost the generalization mechanism of KANs, we propose Lipschitz-Enhanced KANs (LipKANs) by integrating the Lip layers and pioneering the L1.5-regularization, contributing to tighter generalization bounds. Empirical experiments validate that the proposed LipKANs enhance the generalization mechanism of KANs when modeling complex distributions. We hope our theoretical insights and proposed LipKANs lay a foundation for the future development of KANs.

This work presents the first theoretically justified simultaneous inference framework for off-policy evaluation (OPE). In contrast to existing methods that focus on point estimates or pointwise confidence intervals (CIs), the new framework quantifies global uncertainty across an infinite or continuous initial state space, offering valid inference over the entire state space.

In many engineering applications, a single high-fidelity model produces multiple quantities of interest (QoIs) under the same input parameters, e.g. finite element models of complex physical systems. To alleviate the high computational cost of direct model evaluations, surrogate models are widely used to construct efficient approximations of model responses. Naturally, the accuracy of surrogates strongly depends on the quality of the experimental design (ED). However, a single ED may not provide an adequate representation for all outputs simultaneously, especially when different outputs exhibit varying sensitivities to the input variables. A straightforward solution is to perform separate sampling for each output, but this results in increased sampling complexity and computational cost. From a statistical perspective, such an approach also ignores potential correlations among all outputs and may compromise data consistency. To address this issue, an adaptive sequential sampling method for constructing polynomial chaos expansion surrogate models is generalized for vector valued QoIs. The method sequentially selects new samples from a candidate pool based on their local contribution to the output variance, while balancing distance-based exploration of the input space and exploitation of aggregated variance information across all outputs. Its performance is compared with non-sequential Latin Hypercube Sampling through several numerical examples from engineering problems. Numerical results demonstrate that the proposed strategy improves both surrogate accuracy and stability, and provides a more reliable estimation of second-order statistics.

Physics-informed machine learning is gaining significant traction for enhancing statistical performance and sample efficiency through the integration of physical knowledge. However, current theoretical analyses often presume complete prior knowledge in non-hybrid settings, overlooking the crucial integration of observational data, and are frequently limited to linear systems, unlike the prevalent nonlinear nature of many real-world applications. To address these limitations, we introduce a unified residual form that unifies collocation and variational methods, enabling the incorporation of incomplete and complex physical constraints in hybrid learning settings. Within this formulation, we establish that the generalization performance of physics-informed regression in such hybrid settings is governed by the dimension of the affine variety associated with the physical constraint, rather than by the number of parameters. This enables a unified analysis that is applicable to both linear and nonlinear equations. We also present a method to approximate this dimension and provide experimental validation of our theoretical findings.