Deep Conditional Generative Learning: Model and Error Analysis

We introduce an Ordinary Differential Equation (ODE) based deep generative method for learning a conditional distribution, named the Conditional Follmer Flow. Starting from a standard Gaussian distribution, the proposed flow could efficiently transform it into the target conditional distribution at time 1. For effective implementation, we discretize the flow with Euler's method where we estimate the velocity field nonparametrically using a deep neural network. Furthermore, we derive a non-asymptotic convergence rate in the Wasserstein distance between the distribution of the learned samples and the target distribution, providing the first comprehensive end-to-end error analysis for conditional distribution learning via ODE flow. Our numerical experiments showcase its effectiveness across a range of scenarios, from standard nonparametric conditional density estimation problems to more intricate challenges involving image data, illustrating its superiority over various existing conditional density estimation methods.

Semi-Supervised Deep Sobolev Regression: Estimation, Variable Selection and Beyond

We propose SDORE, a semi-supervised deep Sobolev regressor, for the nonparametric estimation of the underlying regression function and its gradient. SDORE employs deep neural networks to minimize empirical risk with gradient norm regularization, allowing computation of the gradient norm on unlabeled data. We conduct a comprehensive analysis of the convergence rates of SDORE and establish a minimax optimal rate for the regression function. Crucially, we also derive a convergence rate for the associated plug-in gradient estimator, even in the presence of significant domain shift. These theoretical findings offer valuable prior guidance for selecting regularization parameters and determining the size of the neural network, while showcasing the provable advantage of leveraging unlabeled data in semi-supervised learning. To the best of our knowledge, SDORE is the first provable neural network-based approach that simultaneously estimates the regression function and its gradient, with diverse applications including nonparametric variable selection and inverse problems. The effectiveness of SDORE is validated through an extensive range of numerical simulations and real data analysis.

Provable Advantage of Parameterized Quantum Circuit in Function Approximation

Understanding the power of parameterized quantum circuits (PQCs) in accomplishing machine learning tasks is one of the most important questions in quantum machine learning. In this paper, we analyze the expressivity of PQCs through the lens of function approximation. Previously established universal approximation theorems for PQCs are mainly nonconstructive, leading us to the following question: How large do the PQCs need to be to approximate the target function up to a given error? We exhibit explicit constructions of data re-uploading PQCs for approximating continuous and smooth functions and establish quantitative approximation error bounds in terms of the width, the depth and the number of trainable parameters of the PQCs. To achieve this, we utilize techniques from quantum signal processing and linear combinations of unitaries to construct PQCs that implement multivariate polynomials. We implement global and local approximation techniques using Bernstein polynomials and local Taylor expansion and analyze their performances in the quantum setting. We also compare our proposed PQCs to nearly optimal deep neural networks in approximating high-dimensional smooth functions, showing that the ratio between model sizes of PQC and deep neural networks is exponentially small with respect to the input dimension. This suggests a potentially novel avenue for showcasing quantum advantages in quantum machine learning.

Current density impedance imaging with PINNs

In this paper, we introduce CDII-PINNs, a computationally efficient method for solving CDII using PINNs in the framework of Tikhonov regularization. This method constructs a physics-informed loss function by merging the regularized least-squares output functional with an underlying differential equation, which describes the relationship between the conductivity and voltage. A pair of neural networks representing the conductivity and voltage, respectively, are coupled by this loss function. Then, minimizing the loss function provides a reconstruction. A rigorous theoretical guarantee is provided. We give an error analysis for CDII-PINNs and establish a convergence rate, based on prior selected neural network parameters in terms of the number of samples. The numerical simulations demonstrate that CDII-PINNs are efficient, accurate and robust to noise levels ranging from $1\%$ to $20\%$.

GAS: A Gaussian Mixture Distribution-Based Adaptive Sampling Method for PINNs

With the recent study of deep learning in scientific computation, the Physics-Informed Neural Networks (PINNs) method has drawn widespread attention for solving Partial Differential Equations (PDEs). Compared to traditional methods, PINNs can efficiently handle high-dimensional problems, but the accuracy is relatively low, especially for highly irregular problems. Inspired by the idea of adaptive finite element methods and incremental learning, we propose GAS, a Gaussian mixture distribution-based adaptive sampling method for PINNs. During the training procedure, GAS uses the current residual information to generate a Gaussian mixture distribution for the sampling of additional points, which are then trained together with historical data to speed up the convergence of the loss and achieve higher accuracy. Several numerical simulations on 2D and 10D problems show that GAS is a promising method that achieves state-of-the-art accuracy among deep solvers, while being comparable with traditional numerical solvers.