arXiv subjects

Zheyuan Hu

Publications and source records attributed to Zheyuan Hu.

35 records · Page 2Linked to original sources

arXiv2024

Score-fPINN: Fractional Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck-Levy Equations

We introduce an innovative approach for solving high-dimensional Fokker-Planck-Lévy (FPL) equations in modeling non-Brownian processes across disciplines such as physics, finance, and ecology. We utilize a fractional score function and Physical-informed neural networks (PINN) to lift the curse of dimensionality (CoD) and alleviate numerical overflow from exponentially decaying solutions with dimensions. The introduction of a fractional score function allows us to transform the FPL equation into a second-order partial differential equation without fractional Laplacian and thus can be readily solved with standard physics-informed neural networks (PINNs). We propose two methods to obtain a fractional score function: fractional score matching (FSM) and score-fPINN for fitting the fractional score function. While FSM is more cost-effective, it relies on known conditional distributions. On the other hand, score-fPINN is independent of specific stochastic differential equations (SDEs) but requires evaluating the PINN model's derivatives, which may be more costly. We conduct our experiments on various SDEs and demonstrate numerical stability and effectiveness of our method in dealing with high-dimensional problems, marking a significant advancement in addressing the CoD in FPL equations.

cs.LG↗

arXiv2024

Tackling the Curse of Dimensionality in Fractional and Tempered Fractional PDEs with Physics-Informed Neural Networks

Fractional and tempered fractional partial differential equations (PDEs) are effective models of long-range interactions, anomalous diffusion, and non-local effects. Traditional numerical methods for these problems are mesh-based, thus struggling with the curse of dimensionality (CoD). Physics-informed neural networks (PINNs) offer a promising solution due to their universal approximation, generalization ability, and mesh-free training. In principle, Monte Carlo fractional PINN (MC-fPINN) estimates fractional derivatives using Monte Carlo methods and thus could lift CoD. However, this may cause significant variance and errors, hence affecting convergence; in addition, MC-fPINN is sensitive to hyperparameters. In general, numerical methods and specifically PINNs for tempered fractional PDEs are under-developed. Herein, we extend MC-fPINN to tempered fractional PDEs to address these issues, resulting in the Monte Carlo tempered fractional PINN (MC-tfPINN). To reduce possible high variance and errors from Monte Carlo sampling, we replace the one-dimensional (1D) Monte Carlo with 1D Gaussian quadrature, applicable to both MC-fPINN and MC-tfPINN. We validate our methods on various forward and inverse problems of fractional and tempered fractional PDEs, scaling up to 100,000 dimensions. Our improved MC-fPINN/MC-tfPINN using quadrature consistently outperforms the original versions in accuracy and convergence speed in very high dimensions.

math.NA↗

arXiv2024

Tackling the Curse of Dimensionality with Physics-Informed Neural Networks

The curse-of-dimensionality taxes computational resources heavily with exponentially increasing computational cost as the dimension increases. This poses great challenges in solving high-dimensional PDEs, as Richard E. Bellman first pointed out over 60 years ago. While there has been some recent success in solving numerically partial differential equations (PDEs) in high dimensions, such computations are prohibitively expensive, and true scaling of general nonlinear PDEs to high dimensions has never been achieved. We develop a new method of scaling up physics-informed neural networks (PINNs) to solve arbitrary high-dimensional PDEs. The new method, called Stochastic Dimension Gradient Descent (SDGD), decomposes a gradient of PDEs into pieces corresponding to different dimensions and randomly samples a subset of these dimensional pieces in each iteration of training PINNs. We prove theoretically the convergence and other desired properties of the proposed method. We demonstrate in various diverse tests that the proposed method can solve many notoriously hard high-dimensional PDEs, including the Hamilton-Jacobi-Bellman (HJB) and the Schrödinger equations in tens of thousands of dimensions very fast on a single GPU using the PINNs mesh-free approach. Notably, we solve nonlinear PDEs with nontrivial, anisotropic, and inseparable solutions in 100,000 effective dimensions in 12 hours on a single GPU using SDGD with PINNs. Since SDGD is a general training methodology of PINNs, it can be applied to any current and future variants of PINNs to scale them up for arbitrary high-dimensional PDEs.

cs.LG↗

arXiv2024

Yell At Your Robot: Improving On-the-Fly from Language Corrections

Hierarchical policies that combine language and low-level control have been shown to perform impressively long-horizon robotic tasks, by leveraging either zero-shot high-level planners like pretrained language and vision-language models (LLMs/VLMs) or models trained on annotated robotic demonstrations. However, for complex and dexterous skills, attaining high success rates on long-horizon tasks still represents a major challenge -- the longer the task is, the more likely it is that some stage will fail. Can humans help the robot to continuously improve its long-horizon task performance through intuitive and natural feedback? In this paper, we make the following observation: high-level policies that index into sufficiently rich and expressive low-level language-conditioned skills can be readily supervised with human feedback in the form of language corrections. We show that even fine-grained corrections, such as small movements ("move a bit to the left"), can be effectively incorporated into high-level policies, and that such corrections can be readily obtained from humans observing the robot and making occasional suggestions. This framework enables robots not only to rapidly adapt to real-time language feedback, but also incorporate this feedback into an iterative training scheme that improves the high-level policy's ability to correct errors in both low-level execution and high-level decision-making purely from verbal feedback. Our evaluation on real hardware shows that this leads to significant performance improvement in long-horizon, dexterous manipulation tasks without the need for any additional teleoperation. Videos and code are available at https://yay-robot.github.io/.

cs.RO↗

arXiv2024

Hutchinson Trace Estimation for High-Dimensional and High-Order Physics-Informed Neural Networks

Physics-Informed Neural Networks (PINNs) have proven effective in solving partial differential equations (PDEs), especially when some data are available by seamlessly blending data and physics. However, extending PINNs to high-dimensional and even high-order PDEs encounters significant challenges due to the computational cost associated with automatic differentiation in the residual loss. Herein, we address the limitations of PINNs in handling high-dimensional and high-order PDEs by introducing Hutchinson Trace Estimation (HTE). Starting with the second-order high-dimensional PDEs ubiquitous in scientific computing, HTE transforms the calculation of the entire Hessian matrix into a Hessian vector product (HVP). This approach alleviates the computational bottleneck via Taylor-mode automatic differentiation and significantly reduces memory consumption from the Hessian matrix to HVP. We further showcase HTE's convergence to the original PINN loss and its unbiased behavior under specific conditions. Comparisons with Stochastic Dimension Gradient Descent (SDGD) highlight the distinct advantages of HTE, particularly in scenarios with significant variance among dimensions. We further extend HTE to higher-order and higher-dimensional PDEs, specifically addressing the biharmonic equation. By employing tensor-vector products (TVP), HTE efficiently computes the colossal tensor associated with the fourth-order high-dimensional biharmonic equation, saving memory and enabling rapid computation. The effectiveness of HTE is illustrated through experimental setups, demonstrating comparable convergence rates with SDGD under memory and speed constraints. Additionally, HTE proves valuable in accelerating the Gradient-Enhanced PINN (gPINN) version as well as the Biharmonic equation. Overall, HTE opens up a new capability in scientific machine learning for tackling high-order and high-dimensional PDEs.

cs.LG↗

arXiv2024

Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck Equations

The Fokker-Planck (FP) equation is a foundational PDE in stochastic processes. However, curse of dimensionality (CoD) poses challenge when dealing with high-dimensional FP PDEs. Although Monte Carlo and vanilla Physics-Informed Neural Networks (PINNs) have shown the potential to tackle CoD, both methods exhibit numerical errors in high dimensions when dealing with the probability density function (PDF) associated with Brownian motion. The point-wise PDF values tend to decrease exponentially as dimension increases, surpassing the precision of numerical simulations and resulting in substantial errors. Moreover, due to its massive sampling, Monte Carlo fails to offer fast sampling. Modeling the logarithm likelihood (LL) via vanilla PINNs transforms the FP equation into a difficult HJB equation, whose error grows rapidly with dimension. To this end, we propose a novel approach utilizing a score-based solver to fit the score function in SDEs. The score function, defined as the gradient of the LL, plays a fundamental role in inferring LL and PDF and enables fast SDE sampling. Three fitting methods, Score Matching (SM), Sliced SM (SSM), and Score-PINN, are introduced. The proposed score-based SDE solver operates in two stages: first, employing SM, SSM, or Score-PINN to acquire the score; and second, solving the LL via an ODE using the obtained score. Comparative evaluations across these methods showcase varying trade-offs. The proposed method is evaluated across diverse SDEs, including anisotropic OU processes, geometric Brownian, and Brownian with varying eigenspace. We also test various distributions, including Gaussian, Log-normal, Laplace, and Cauchy. The numerical results demonstrate the score-based SDE solver's stability, speed, and performance across different settings, solidifying its potential as a solution to CoD for high-dimensional FP equations.

cs.LG↗

arXiv2023

Bias-Variance Trade-off in Physics-Informed Neural Networks with Randomized Smoothing for High-Dimensional PDEs

While physics-informed neural networks (PINNs) have been proven effective for low-dimensional partial differential equations (PDEs), the computational cost remains a hurdle in high-dimensional scenarios. This is particularly pronounced when computing high-order and high-dimensional derivatives in the physics-informed loss. Randomized Smoothing PINN (RS-PINN) introduces Gaussian noise for stochastic smoothing of the original neural net model, enabling Monte Carlo methods for derivative approximation, eliminating the need for costly auto-differentiation. Despite its computational efficiency in high dimensions, RS-PINN introduces biases in both loss and gradients, negatively impacting convergence, especially when coupled with stochastic gradient descent (SGD). We present a comprehensive analysis of biases in RS-PINN, attributing them to the nonlinearity of the Mean Squared Error (MSE) loss and the PDE nonlinearity. We propose tailored bias correction techniques based on the order of PDE nonlinearity. The unbiased RS-PINN allows for a detailed examination of its pros and cons compared to the biased version. Specifically, the biased version has a lower variance and runs faster than the unbiased version, but it is less accurate due to the bias. To optimize the bias-variance trade-off, we combine the two approaches in a hybrid method that balances the rapid convergence of the biased version with the high accuracy of the unbiased version. In addition, we present an enhanced implementation of RS-PINN. Extensive experiments on diverse high-dimensional PDEs, including Fokker-Planck, HJB, viscous Burgers', Allen-Cahn, and Sine-Gordon equations, illustrate the bias-variance trade-off and highlight the effectiveness of the hybrid RS-PINN. Empirical guidelines are provided for selecting biased, unbiased, or hybrid versions, depending on the dimensionality and nonlinearity of the specific PDE problem.

cs.LG↗

arXiv2023

Augmented Physics-Informed Neural Networks (APINNs): A gating network-based soft domain decomposition methodology

In this paper, we propose the augmented physics-informed neural network (APINN), which adopts soft and trainable domain decomposition and flexible parameter sharing to further improve the extended PINN (XPINN) as well as the vanilla PINN methods. In particular, a trainable gate network is employed to mimic the hard decomposition of XPINN, which can be flexibly fine-tuned for discovering a potentially better partition. It weight-averages several sub-nets as the output of APINN. APINN does not require complex interface conditions, and its sub-nets can take advantage of all training samples rather than just part of the training data in their subdomains. Lastly, each sub-net shares part of the common parameters to capture the similar components in each decomposed function. Furthermore, following the PINN generalization theory in Hu et al. [2021], we show that APINN can improve generalization by proper gate network initialization and general domain & function decomposition. Extensive experiments on different types of PDEs demonstrate how APINN improves the PINN and XPINN methods. Specifically, we present examples where XPINN performs similarly to or worse than PINN, so that APINN can significantly improve both. We also show cases where XPINN is already better than PINN, so APINN can still slightly improve XPINN. Furthermore, we visualize the optimized gating networks and their optimization trajectories, and connect them with their performance, which helps discover the possibly optimal decomposition. Interestingly, if initialized by different decomposition, the performances of corresponding APINNs can differ drastically. This, in turn, shows the potential to design an optimal domain decomposition for the differential equation problem under consideration.

cs.LG↗

arXiv2023

REBOOT: Reuse Data for Bootstrapping Efficient Real-World Dexterous Manipulation

Dexterous manipulation tasks involving contact-rich interactions pose a significant challenge for both model-based control systems and imitation learning algorithms. The complexity arises from the need for multi-fingered robotic hands to dynamically establish and break contacts, balance non-prehensile forces, and control large degrees of freedom. Reinforcement learning (RL) offers a promising approach due to its general applicability and capacity to autonomously acquire optimal manipulation strategies. However, its real-world application is often hindered by the necessity to generate a large number of samples, reset the environment, and obtain reward signals. In this work, we introduce an efficient system for learning dexterous manipulation skills with RL to alleviate these challenges. The main idea of our approach is the integration of recent advances in sample-efficient RL and replay buffer bootstrapping. This combination allows us to utilize data from different tasks or objects as a starting point for training new tasks, significantly improving learning efficiency. Additionally, our system completes the real-world training cycle by incorporating learned resets via an imitation-based pickup policy as well as learned reward functions, eliminating the need for manual resets and reward engineering. We demonstrate the benefits of reusing past data as replay buffer initialization for new tasks, for instance, the fast acquisition of intricate manipulation skills in the real world on a four-fingered robotic hand. (Videos: https://sites.google.com/view/reboot-dexterous)

cs.LG↗

arXiv2023

D4FT: A Deep Learning Approach to Kohn-Sham Density Functional Theory

Kohn-Sham Density Functional Theory (KS-DFT) has been traditionally solved by the Self-Consistent Field (SCF) method. Behind the SCF loop is the physics intuition of solving a system of non-interactive single-electron wave functions under an effective potential. In this work, we propose a deep learning approach to KS-DFT. First, in contrast to the conventional SCF loop, we propose to directly minimize the total energy by reparameterizing the orthogonal constraint as a feed-forward computation. We prove that such an approach has the same expressivity as the SCF method, yet reduces the computational complexity from O(N^4) to O(N^3). Second, the numerical integration which involves a summation over the quadrature grids can be amortized to the optimization steps. At each step, stochastic gradient descent (SGD) is performed with a sampled minibatch of the grids. Extensive experiments are carried out to demonstrate the advantage of our approach in terms of efficiency and stability. In addition, we show that our approach enables us to explore more complex neural-based wave functions.

cs.LG↗

arXiv2022

Dexterous Manipulation from Images: Autonomous Real-World RL via Substep Guidance

Complex and contact-rich robotic manipulation tasks, particularly those that involve multi-fingered hands and underactuated object manipulation, present a significant challenge to any control method. Methods based on reinforcement learning offer an appealing choice for such settings, as they can enable robots to learn to delicately balance contact forces and dexterously reposition objects without strong modeling assumptions. However, running reinforcement learning on real-world dexterous manipulation systems often requires significant manual engineering. This negates the benefits of autonomous data collection and ease of use that reinforcement learning should in principle provide. In this paper, we describe a system for vision-based dexterous manipulation that provides a "programming-free" approach for users to define new tasks and enable robots with complex multi-fingered hands to learn to perform them through interaction. The core principle underlying our system is that, in a vision-based setting, users should be able to provide high-level intermediate supervision that circumvents challenges in teleoperation or kinesthetic teaching which allow a robot to not only learn a task efficiently but also to autonomously practice. Our system includes a framework for users to define a final task and intermediate sub-tasks with image examples, a reinforcement learning procedure that learns the task autonomously without interventions, and experimental results with a four-finger robotic hand learning multi-stage object manipulation tasks directly in the real world, without simulation, manual modeling, or reward engineering.

cs.LG↗

arXiv2022

When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?

Physics-informed neural networks (PINNs) have become a popular choice for solving high-dimensional partial differential equations (PDEs) due to their excellent approximation power and generalization ability. Recently, Extended PINNs (XPINNs) based on domain decomposition methods have attracted considerable attention due to their effectiveness in modeling multiscale and multiphysics problems and their parallelization. However, theoretical understanding on their convergence and generalization properties remains unexplored. In this study, we take an initial step towards understanding how and when XPINNs outperform PINNs. Specifically, for general multi-layer PINNs and XPINNs, we first provide a prior generalization bound via the complexity of the target functions in the PDE problem, and a posterior generalization bound via the posterior matrix norms of the networks after optimization. Moreover, based on our bounds, we analyze the conditions under which XPINNs improve generalization. Concretely, our theory shows that the key building block of XPINN, namely the domain decomposition, introduces a tradeoff for generalization. On the one hand, XPINNs decompose the complex PDE solution into several simple parts, which decreases the complexity needed to learn each part and boosts generalization. On the other hand, decomposition leads to less training data being available in each subdomain, and hence such model is typically prone to overfitting and may become less generalizable. Empirically, we choose five PDEs to show when XPINNs perform better than, similar to, or worse than PINNs, hence demonstrating and justifying our new theory.

cs.LG↗

arXiv2022

Enhancing Collaborative Filtering Recommender with Prompt-Based Sentiment Analysis

Collaborative Filtering(CF) recommender is a crucial application in the online market and ecommerce. However, CF recommender has been proven to suffer from persistent problems related to sparsity of the user rating that will further lead to a cold-start issue. Existing methods address the data sparsity issue by applying token-level sentiment analysis that translate text review into sentiment scores as a complement of the user rating. In this paper, we attempt to optimize the sentiment analysis with advanced NLP models including BERT and RoBERTa, and experiment on whether the CF recommender has been further enhanced. We build the recommenders on the Amazon US Reviews dataset, and tune the pretrained BERT and RoBERTa with the traditional fine-tuned paradigm as well as the new prompt-based learning paradigm. Experimental result shows that the recommender enhanced with the sentiment ratings predicted by the fine-tuned RoBERTa has the best performance, and achieved 30.7% overall gain by comparing MAP, NDCG and precision at K to the baseline recommender. Prompt-based learning paradigm, although superior to traditional fine-tune paradigm in pure sentiment analysis, fail to further improve the CF recommender.

cs.IR↗

arXiv2022

TNT: Vision Transformer for Turbulence Simulations

Turbulence is notoriously difficult to model due to its multi-scale nature and sensitivity to small perturbations. Classical solvers of turbulence simulation generally operate on finer grids and are computationally inefficient. In this paper, we propose the Turbulence Neural Transformer (TNT), which is a learned simulator based on the transformer architecture, to predict turbulent dynamics on coarsened grids. TNT extends the positional embeddings of vanilla transformers to a spatiotemporal setting to learn the representation in the 3D time-series domain, and applies Temporal Mutual Self-Attention (TMSA), which captures adjacent dependencies, to extract deep and dynamic features. TNT is capable of generating comparatively long-range predictions stably and accurately, and we show that TNT outperforms the state-of-the-art U-net simulator on several metrics. We also test the model performance with different components removed and evaluate robustness to different initial conditions. Although more experiments are needed, we conclude that TNT has great potential to outperform existing solvers and generalize to additional simulation datasets.

physics.flu-dyn↗

arXiv2021

Kernelized Heterogeneous Risk Minimization

The ability to generalize under distributional shifts is essential to reliable machine learning, while models optimized with empirical risk minimization usually fail on non-$i.i.d$ testing data. Recently, invariant learning methods for out-of-distribution (OOD) generalization propose to find causally invariant relationships with multi-environments. However, modern datasets are frequently multi-sourced without explicit source labels, rendering many invariant learning methods inapplicable. In this paper, we propose Kernelized Heterogeneous Risk Minimization (KerHRM) algorithm, which achieves both the latent heterogeneity exploration and invariant learning in kernel space, and then gives feedback to the original neural network by appointing invariant gradient direction. We theoretically justify our algorithm and empirically validate the effectiveness of our algorithm with extensive experiments.

cs.LG↗

arXiv2021

Heterogeneous Risk Minimization

Machine learning algorithms with empirical risk minimization usually suffer from poor generalization performance due to the greedy exploitation of correlations among the training data, which are not stable under distributional shifts. Recently, some invariant learning methods for out-of-distribution (OOD) generalization have been proposed by leveraging multiple training environments to find invariant relationships. However, modern datasets are frequently assembled by merging data from multiple sources without explicit source labels. The resultant unobserved heterogeneity renders many invariant learning methods inapplicable. In this paper, we propose Heterogeneous Risk Minimization (HRM) framework to achieve joint learning of latent heterogeneity among the data and invariant relationship, which leads to stable prediction despite distributional shifts. We theoretically characterize the roles of the environment labels in invariant learning and justify our newly proposed HRM framework. Extensive experimental results validate the effectiveness of our HRM framework.

cs.LG↗

arXiv2020

ASGN: An Active Semi-supervised Graph Neural Network for Molecular Property Prediction

Molecular property prediction (e.g., energy) is an essential problem in chemistry and biology. Unfortunately, many supervised learning methods usually suffer from the problem of scarce labeled molecules in the chemical space, where such property labels are generally obtained by Density Functional Theory (DFT) calculation which is extremely computational costly. An effective solution is to incorporate the unlabeled molecules in a semi-supervised fashion. However, learning semi-supervised representation for large amounts of molecules is challenging, including the joint representation issue of both molecular essence and structure, the conflict between representation and property leaning. Here we propose a novel framework called Active Semi-supervised Graph Neural Network (ASGN) by incorporating both labeled and unlabeled molecules. Specifically, ASGN adopts a teacher-student framework. In the teacher model, we propose a novel semi-supervised learning method to learn general representation that jointly exploits information from molecular structure and molecular distribution. Then in the student model, we target at property prediction task to deal with the learning loss conflict. At last, we proposed a novel active learning strategy in terms of molecular diversities to select informative data during the whole framework learning. We conduct extensive experiments on several public datasets. Experimental results show the remarkable performance of our ASGN framework.

cs.LG↗