Physics-informed polynomial chaos expansions (PC$^2$) provide an efficient physically constrained surrogate modeling framework by embedding governing equations and other physical constraints into the standard data-driven polynomial chaos expansions (PCE) and solving via the Karush-Kuhn-Tucker (KKT) conditions. This approach improves the physical interpretability of surrogate models while achieving high computational efficiency and accuracy. However, the performance and efficiency of PC$^2$ can still be degraded with high-dimensional parameter spaces, limited data availability, or unrepresentative training data. To address this problem, this study explores two complementary enhancements to the PC$^2$ framework. First, a numerically efficient constrained optimization solver, straightforward updating of Lagrange multipliers (SULM), is adopted as an alternative to the conventional KKT solver. The SULM method significantly reduces computational cost when solving physically constrained problems with high-dimensionality and derivative boundary conditions that require a large number of virtual points. Second, a D-optimal sampling strategy is utilized to select informative virtual points to improve the stability and achieve the balance of accuracy and efficiency of the PC$^2$. The proposed methods are integrated into the PC$^2$ framework and evaluated through numerical examples of representative physical systems governed by ordinary or partial differential equations. The results demonstrate that the enhanced PC$^2$ has better comprehensive capability than standard PC$^2$, and is well-suited for high-dimensional uncertainty quantification tasks.

Operator learning (OL) has emerged as a powerful tool in scientific machine learning (SciML) for approximating mappings between infinite-dimensional functional spaces. One of its main applications is learning the solution operator of partial differential equations (PDEs). While much of the progress in this area has been driven by deep neural network-based approaches such as Deep Operator Networks (DeepONet) and Fourier Neural Operator (FNO), recent work has begun to explore traditional machine learning methods for OL. In this work, we introduce polynomial chaos expansion (PCE) as an OL method. PCE has been widely used for uncertainty quantification (UQ) and has recently gained attention in the context of SciML. For OL, we establish a mathematical framework that enables PCE to approximate operators in both purely data-driven and physics-informed settings. The proposed framework reduces the task of learning the operator to solving a system of equations for the PCE coefficients. Moreover, the framework provides UQ by simply post-processing the PCE coefficients, without any additional computational cost. We apply the proposed method to a diverse set of PDE problems to demonstrate its capabilities. Numerical results demonstrate the strong performance of the proposed method in both OL and UQ tasks, achieving excellent numerical accuracy and computational efficiency.

We present and discuss applications of the category of probabilistic morphisms, initially developed in \cite{Le2023}, as well as some geometric methods to several classes of problems in statistical, machine and manifold learning which shall be, along with many other topics, considered in depth in the forthcoming book \cite{LMPT2024}.

In the realm of cross-modal retrieval, seamlessly integrating diverse modalities within multimedia remains a formidable challenge, especially given the complexities introduced by noisy correspondence learning (NCL). Such noise often stems from mismatched data pairs, which is a significant obstacle distinct from traditional noisy labels. This paper introduces Pseudo-Classification based Pseudo-Captioning (PC$^2$) framework to address this challenge. PC$^2$ offers a threefold strategy: firstly, it establishes an auxiliary "pseudo-classification" task that interprets captions as categorical labels, steering the model to learn image-text semantic similarity through a non-contrastive mechanism. Secondly, unlike prevailing margin-based techniques, capitalizing on PC$^2$'s pseudo-classification capability, we generate pseudo-captions to provide more informative and tangible supervision for each mismatched pair. Thirdly, the oscillation of pseudo-classification is borrowed to assistant the correction of correspondence. In addition to technical contributions, we develop a realistic NCL dataset called Noise of Web (NoW), which could be a new powerful NCL benchmark where noise exists naturally. Empirical evaluations of PC$^2$ showcase marked improvements over existing state-of-the-art robust cross-modal retrieval techniques on both simulated and realistic datasets with various NCL settings. The contributed dataset and source code are released at https://github.com/alipay/PC2-NoiseofWeb.

We present Bayesian Diffusion Models (BDM), a prediction algorithm that performs effective Bayesian inference by tightly coupling the top-down (prior) information with the bottom-up (data-driven) procedure via joint diffusion processes. We show the effectiveness of BDM on the 3D shape reconstruction task. Compared to prototypical deep learning data-driven approaches trained on paired (supervised) data-labels (e.g. image-point clouds) datasets, our BDM brings in rich prior information from standalone labels (e.g. point clouds) to improve the bottom-up 3D reconstruction. As opposed to the standard Bayesian frameworks where explicit prior and likelihood are required for the inference, BDM performs seamless information fusion via coupled diffusion processes with learned gradient computation networks. The specialty of our BDM lies in its capability to engage the active and effective information exchange and fusion of the top-down and bottom-up processes where each itself is a diffusion process. We demonstrate state-of-the-art results on both synthetic and real-world benchmarks for 3D shape reconstruction.

We present a novel physics-constrained polynomial chaos expansion as a surrogate modeling method capable of performing both scientific machine learning (SciML) and uncertainty quantification (UQ) tasks. The proposed method possesses a unique capability: it seamlessly integrates SciML into UQ and vice versa, which allows it to quantify the uncertainties in SciML tasks effectively and leverage SciML for improved uncertainty assessment during UQ-related tasks. The proposed surrogate model can effectively incorporate a variety of physical constraints, such as governing partial differential equations (PDEs) with associated initial and boundary conditions constraints, inequality-type constraints (e.g., monotonicity, convexity, non-negativity, among others), and additional a priori information in the training process to supplement limited data. This ensures physically realistic predictions and significantly reduces the need for expensive computational model evaluations to train the surrogate model. Furthermore, the proposed method has a built-in uncertainty quantification (UQ) feature to efficiently estimate output uncertainties. To demonstrate the effectiveness of the proposed method, we apply it to a diverse set of problems, including linear/non-linear PDEs with deterministic and stochastic parameters, data-driven surrogate modeling of a complex physical system, and UQ of a stochastic system with parameters modeled as random fields.

Surrogate modeling of costly mathematical models representing physical systems is challenging since it is typically not possible to create a large experimental design. Thus, it is beneficial to constrain the approximation to adhere to the known physics of the model. This paper presents a novel methodology for the construction of physics-informed polynomial chaos expansions (PCE) that combines the conventional experimental design with additional constraints from the physics of the model. Physical constraints investigated in this paper are represented by a set of differential equations and specified boundary conditions. A computationally efficient means for construction of physically constrained PCE is proposed and compared to standard sparse PCE. It is shown that the proposed algorithms lead to superior accuracy of the approximation and does not add significant computational burden. Although the main purpose of the proposed method lies in combining data and physical constraints, we show that physically constrained PCEs can be constructed from differential equations and boundary conditions alone without requiring evaluations of the original model. We further show that the constrained PCEs can be easily applied for uncertainty quantification through analytical post-processing of a reduced PCE filtering out the influence of all deterministic space-time variables. Several deterministic examples of increasing complexity are provided and the proposed method is applied for uncertainty quantification.

Recurrent neural network-based reinforcement learning systems are capable of complex motor control tasks such as locomotion and manipulation, however, much of their underlying mechanisms still remain difficult to interpret. Our aim is to leverage computational neuroscience methodologies to understanding the population-level activity of robust robot locomotion controllers. Our investigation begins by analyzing topological structure, discovering that fragile controllers have a higher number of fixed points with unstable directions, resulting in poorer balance when instructed to stand in place. Next, we analyze the forced response of the system by applying targeted neural perturbations along directions of dominant population-level activity. We find evidence that recurrent state dynamics are structured and low-dimensional during walking, which aligns with primate studies. Additionally, when recurrent states are perturbed to zero, fragile agents continue to walk, which is indicative of a stronger reliance on sensory input and weaker recurrence.

Multi-server queueing systems are widely used models for job scheduling in machine learning, wireless networks, crowdsourcing, and healthcare systems. This paper considers a multi-server system with multiple servers and multiple types of jobs, where different job types require different amounts of processing time at different servers. The goal is to schedule jobs on servers without knowing the statistics of the processing times. To fully utilize the processing power of the servers, it is known that one has to at least learn the service rates of different job types on different servers. Prior works on this topic decouple the learning and scheduling phases which leads to either excessive exploration or extremely large job delays. We propose a new algorithm, which combines the MaxWeight scheduling policy with discounted upper confidence bound (UCB), to simultaneously learn the statistics and schedule jobs to servers. We prove that under our algorithm the asymptotic average queue length is bounded by one divided by the traffic slackness, which is order-wise optimal. We also obtain an exponentially decaying probability tail bound for any-time queue length. These results hold for both stationary and nonstationary service rates. Simulations confirm that the delay performance of our algorithm is several orders of magnitude better than previously proposed algorithms.

Reconstructing the 3D shape of an object from a single RGB image is a long-standing and highly challenging problem in computer vision. In this paper, we propose a novel method for single-image 3D reconstruction which generates a sparse point cloud via a conditional denoising diffusion process. Our method takes as input a single RGB image along with its camera pose and gradually denoises a set of 3D points, whose positions are initially sampled randomly from a three-dimensional Gaussian distribution, into the shape of an object. The key to our method is a geometrically-consistent conditioning process which we call projection conditioning: at each step in the diffusion process, we project local image features onto the partially-denoised point cloud from the given camera pose. This projection conditioning process enables us to generate high-resolution sparse geometries that are well-aligned with the input image, and can additionally be used to predict point colors after shape reconstruction. Moreover, due to the probabilistic nature of the diffusion process, our method is naturally capable of generating multiple different shapes consistent with a single input image. In contrast to prior work, our approach not only performs well on synthetic benchmarks, but also gives large qualitative improvements on complex real-world data.