arXiv subjects

Tianrong Chen

Publications and source records attributed to Tianrong Chen.

33 records · Page 2Linked to original sources

arXiv2024

React-OT: Optimal Transport for Generating Transition State in Chemical Reactions

Transition states (TSs) are transient structures that are key in understanding reaction mechanisms and designing catalysts but challenging to be captured in experiments. Alternatively, many optimization algorithms have been developed to search for TSs computationally. Yet the cost of these algorithms driven by quantum chemistry methods (usually density functional theory) is still high, posing challenges for their applications in building large reaction networks for reaction exploration. Here we developed React-OT, an optimal transport approach for generating unique TS structures from reactants and products. React-OT generates highly accurate TS structures with a median structural root mean square deviation (RMSD) of 0.053Å and median barrier height error of 1.06 kcal/mol requiring only 0.4 second per reaction. The RMSD and barrier height error is further improved by roughly 25\% through pretraining React-OT on a large reaction dataset obtained with a lower level of theory, GFN2-xTB. We envision that the remarkable accuracy and rapid inference of React-OT will be highly useful when integrated with the current high-throughput TS search workflow. This integration will facilitate the exploration of chemical reactions with unknown mechanisms.

physics.chem-ph↗

arXiv2024

Quantum State Generation with Structure-Preserving Diffusion Model

This article considers the generative modeling of the (mixed) states of quantum systems, and an approach based on denoising diffusion model is proposed. The key contribution is an algorithmic innovation that respects the physical nature of quantum states. More precisely, the commonly used density matrix representation of mixed-state has to be complex-valued Hermitian, positive semi-definite, and trace one. Generic diffusion models, or other generative methods, may not be able to generate data that strictly satisfy these structural constraints, even if all training data do. To develop a machine learning algorithm that has physics hard-wired in, we leverage mirror diffusion and borrow the physical notion of von Neumann entropy to design a new map, for enabling strict structure-preserving generation. Both unconditional generation and conditional generation via classifier-free guidance are experimentally demonstrated efficacious, the latter enabling the design of new quantum states when generated on unseen labels.

quant-ph↗

arXiv2024

Generative Modeling with Phase Stochastic Bridges

Diffusion models (DMs) represent state-of-the-art generative models for continuous inputs. DMs work by constructing a Stochastic Differential Equation (SDE) in the input space (ie, position space), and using a neural network to reverse it. In this work, we introduce a novel generative modeling framework grounded in \textbf{phase space dynamics}, where a phase space is defined as {an augmented space encompassing both position and velocity.} Leveraging insights from Stochastic Optimal Control, we construct a path measure in the phase space that enables efficient sampling. {In contrast to DMs, our framework demonstrates the capability to generate realistic data points at an early stage of dynamics propagation.} This early prediction sets the stage for efficient data generation by leveraging additional velocity information along the trajectory. On standard image generation benchmarks, our model yields favorable performance over baselines in the regime of small Number of Function Evaluations (NFEs). Furthermore, our approach rivals the performance of diffusion models equipped with efficient sampling techniques, underscoring its potential as a new tool generative modeling.

cs.LG↗

arXiv2024

Mirror Diffusion Models for Constrained and Watermarked Generation

Modern successes of diffusion models in learning complex, high-dimensional data distributions are attributed, in part, to their capability to construct diffusion processes with analytic transition kernels and score functions. The tractability results in a simulation-free framework with stable regression losses, from which reversed, generative processes can be learned at scale. However, when data is confined to a constrained set as opposed to a standard Euclidean space, these desirable characteristics appear to be lost based on prior attempts. In this work, we propose Mirror Diffusion Models (MDM), a new class of diffusion models that generate data on convex constrained sets without losing any tractability. This is achieved by learning diffusion processes in a dual space constructed from a mirror map, which, crucially, is a standard Euclidean space. We derive efficient computation of mirror maps for popular constrained sets, such as simplices and $\ell_2$-balls, showing significantly improved performance of MDM over existing methods. For safety and privacy purposes, we also explore constrained sets as a new mechanism to embed invisible but quantitative information (i.e., watermarks) in generated data, for which MDM serves as a compelling approach. Our work brings new algorithmic opportunities for learning tractable diffusion on complex domains. Our code is available at https://github.com/ghliu/mdm

stat.ML↗

arXiv2023

Augmented Bridge Matching

Flow and bridge matching are a novel class of processes which encompass diffusion models. One of the main aspect of their increased flexibility is that these models can interpolate between arbitrary data distributions i.e. they generalize beyond generative modeling and can be applied to learning stochastic (and deterministic) processes of arbitrary transfer tasks between two given distributions. In this paper, we highlight that while flow and bridge matching processes preserve the information of the marginal distributions, they do \emph{not} necessarily preserve the coupling information unless additional, stronger optimality conditions are met. This can be problematic if one aims at preserving the original empirical pairing. We show that a simple modification of the matching process recovers this coupling by augmenting the velocity field (or drift) with the information of the initial sample point. Doing so, we lose the Markovian property of the process but preserve the coupling information between distributions. We illustrate the efficiency of our augmentation in learning mixture of image translation tasks.

cs.LG↗

arXiv2023

Deep Momentum Multi-Marginal Schrödinger Bridge

It is a crucial challenge to reconstruct population dynamics using unlabeled samples from distributions at coarse time intervals. Recent approaches such as flow-based models or Schrödinger Bridge (SB) models have demonstrated appealing performance, yet the inferred sample trajectories either fail to account for the underlying stochasticity or are $\underline{D}$eep $\underline{M}$omentum Multi-Marginal $\underline{S}$chrödinger $\underline{B}$ridge(DMSB), a novel computational framework that learns the smooth measure-valued spline for stochastic systems that satisfy position marginal constraints across time. By tailoring the celebrated Bregman Iteration and extending the Iteration Proportional Fitting to phase space, we manage to handle high-dimensional multi-marginal trajectory inference tasks efficiently. Our algorithm outperforms baselines significantly, as evidenced by experiments for synthetic datasets and a real-world single-cell RNA sequence dataset. Additionally, the proposed approach can reasonably reconstruct the evolution of velocity distribution, from position snapshots only, when there is a ground truth velocity that is nevertheless inaccessible.

stat.ML↗

arXiv2023

Likelihood Training of Schrödinger Bridge using Forward-Backward SDEs Theory

Schrödinger Bridge (SB) is an entropy-regularized optimal transport problem that has received increasing attention in deep generative modeling for its mathematical flexibility compared to the Scored-based Generative Model (SGM). However, it remains unclear whether the optimization principle of SB relates to the modern training of deep generative models, which often rely on constructing log-likelihood objectives.This raises questions on the suitability of SB models as a principled alternative for generative applications. In this work, we present a novel computational framework for likelihood training of SB models grounded on Forward-Backward Stochastic Differential Equations Theory - a mathematical methodology appeared in stochastic optimal control that transforms the optimality condition of SB into a set of SDEs. Crucially, these SDEs can be used to construct the likelihood objectives for SB that, surprisingly, generalizes the ones for SGM as special cases. This leads to a new optimization principle that inherits the same SB optimality yet without losing applications of modern generative training techniques, and we show that the resulting training algorithm achieves comparable results on generating realistic images on MNIST, CelebA, and CIFAR10. Our code is available at https://github.com/ghliu/SB-FBSDE.

stat.ML↗

arXiv2022

Deep Generalized Schrödinger Bridge

Mean-Field Game (MFG) serves as a crucial mathematical framework in modeling the collective behavior of individual agents interacting stochastically with a large population. In this work, we aim at solving a challenging class of MFGs in which the differentiability of these interacting preferences may not be available to the solver, and the population is urged to converge exactly to some desired distribution. These setups are, despite being well-motivated for practical purposes, complicated enough to paralyze most (deep) numerical solvers. Nevertheless, we show that Schrödinger Bridge - as an entropy-regularized optimal transport model - can be generalized to accepting mean-field structures, hence solving these MFGs. This is achieved via the application of Forward-Backward Stochastic Differential Equations theory, which, intriguingly, leads to a computational framework with a similar structure to Temporal Difference learning. As such, it opens up novel algorithmic connections to Deep Reinforcement Learning that we leverage to facilitate practical training. We show that our proposed objective function provides necessary and sufficient conditions to the mean-field problem. Our method, named Deep Generalized Schrödinger Bridge (DeepGSB), not only outperforms prior methods in solving classical population navigation MFGs, but is also capable of solving 1000-dimensional opinion depolarization, setting a new state-of-the-art numerical solver for high-dimensional MFGs. Our code will be made available at https://github.com/ghliu/DeepGSB.

stat.ML↗

arXiv2022

Deep Graphic FBSDEs for Opinion Dynamics Stochastic Control

In this paper, we present a scalable deep learning approach to solve opinion dynamics stochastic optimal control problems with mean field term coupling in the dynamics and cost function. Our approach relies on the probabilistic representation of the solution of the Hamilton-Jacobi-Bellman partial differential equation. Grounded on the nonlinear version of the Feynman-Kac lemma, the solutions of the Hamilton-Jacobi-Bellman partial differential equation are linked to the solution of Forward-Backward Stochastic Differential Equations. These equations can be solved numerically using a novel deep neural network with architecture tailored to the problem in consideration. The resulting algorithm is tested on a polarized opinion consensus experiment. The large-scale (10K) agents experiment validates the scalability and generalizability of our algorithm. The proposed framework opens up the possibility for future applications on extremely large-scale problems.

cs.MA↗

arXiv2021

Second-Order Neural ODE Optimizer

We propose a novel second-order optimization framework for training the emerging deep continuous-time models, specifically the Neural Ordinary Differential Equations (Neural ODEs). Since their training already involves expensive gradient computation by solving a backward ODE, deriving efficient second-order methods becomes highly nontrivial. Nevertheless, inspired by the recent Optimal Control (OC) interpretation of training deep networks, we show that a specific continuous-time OC methodology, called Differential Programming, can be adopted to derive backward ODEs for higher-order derivatives at the same O(1) memory cost. We further explore a low-rank representation of the second-order derivatives and show that it leads to efficient preconditioned updates with the aid of Kronecker-based factorization. The resulting method -- named SNOpt -- converges much faster than first-order baselines in wall-clock time, and the improvement remains consistent across various applications, e.g. image classification, generative flow, and time-series prediction. Our framework also enables direct architecture optimization, such as the integration time of Neural ODEs, with second-order feedback policies, strengthening the OC perspective as a principled tool of analyzing optimization in deep learning. Our code is available at https://github.com/ghliu/snopt.

cs.LG↗

arXiv2021

Dynamic Game Theoretic Neural Optimizer

The connection between training deep neural networks (DNNs) and optimal control theory (OCT) has attracted considerable attention as a principled tool of algorithmic design. Despite few attempts being made, they have been limited to architectures where the layer propagation resembles a Markovian dynamical system. This casts doubts on their flexibility to modern networks that heavily rely on non-Markovian dependencies between layers (e.g. skip connections in residual networks). In this work, we propose a novel dynamic game perspective by viewing each layer as a player in a dynamic game characterized by the DNN itself. Through this lens, different classes of optimizers can be seen as matching different types of Nash equilibria, depending on the implicit information structure of each (p)layer. The resulting method, called Dynamic Game Theoretic Neural Optimizer (DGNOpt), not only generalizes OCT-inspired optimizers to richer network class; it also motivates a new training principle by solving a multi-player cooperative game. DGNOpt shows convergence improvements over existing methods on image classification datasets with residual and inception networks. Our work marries strengths from both OCT and game theory, paving ways to new algorithmic opportunities from robust optimal control and bandit-based optimization.

cs.LG↗

arXiv2021

Large-Scale Multi-Agent Deep FBSDEs

In this paper we present a scalable deep learning framework for finding Markovian Nash Equilibria in multi-agent stochastic games using fictitious play. The motivation is inspired by theoretical analysis of Forward Backward Stochastic Differential Equations (FBSDE) and their implementation in a deep learning setting, which is the source of our algorithm's sample efficiency improvement. By taking advantage of the permutation-invariant property of agents in symmetric games, the scalability and performance is further enhanced significantly. We showcase superior performance of our framework over the state-of-the-art deep fictitious play algorithm on an inter-bank lending/borrowing problem in terms of multiple metrics. More importantly, our approach scales up to 3000 agents in simulation, a scale which, to the best of our knowledge, represents a new state-of-the-art. We also demonstrate the applicability of our framework in robotics on a belief space autonomous racing problem.

cs.AI↗

arXiv2021

DDPNOpt: Differential Dynamic Programming Neural Optimizer

Interpretation of Deep Neural Networks (DNNs) training as an optimal control problem with nonlinear dynamical systems has received considerable attention recently, yet the algorithmic development remains relatively limited. In this work, we make an attempt along this line by reformulating the training procedure from the trajectory optimization perspective. We first show that most widely-used algorithms for training DNNs can be linked to the Differential Dynamic Programming (DDP), a celebrated second-order method rooted in the Approximate Dynamic Programming. In this vein, we propose a new class of optimizer, DDP Neural Optimizer (DDPNOpt), for training feedforward and convolution networks. DDPNOpt features layer-wise feedback policies which improve convergence and reduce sensitivity to hyper-parameter over existing methods. It outperforms other optimal-control inspired training methods in both convergence and complexity, and is competitive against state-of-the-art first and second order methods. We also observe DDPNOpt has surprising benefit in preventing gradient vanishing. Our work opens up new avenues for principled algorithmic design built upon the optimal control theory.

cs.LG↗

arXiv2020

A Differential Game Theoretic Neural Optimizer for Training Residual Networks

Connections between Deep Neural Networks (DNNs) training and optimal control theory has attracted considerable attention as a principled tool of algorithmic design. Differential Dynamic Programming (DDP) neural optimizer is a recently proposed method along this line. Despite its empirical success, the applicability has been limited to feedforward networks and whether such a trajectory-optimization inspired framework can be extended to modern architectures remains unclear. In this work, we derive a generalized DDP optimizer that accepts both residual connections and convolution layers. The resulting optimal control representation admits a game theoretic perspective, in which training residual networks can be interpreted as cooperative trajectory optimization on state-augmented dynamical systems. This Game Theoretic DDP (GT-DDP) optimizer enjoys the same theoretic connection in previous work, yet generates a much complex update rule that better leverages available information during network propagation. Evaluation on image classification datasets (e.g. MNIST and CIFAR100) shows an improvement in training convergence and variance reduction over existing methods. Our approach highlights the benefit gained from architecture-aware optimization.

cs.LG↗

arXiv2019

Deep 2FBSDEs For Systems With Control Multiplicative Noise

We present a deep recurrent neural network architecture to solve a class of stochastic optimal control problems described by fully nonlinear Hamilton Jacobi Bellmanpartial differential equations. Such PDEs arise when one considers stochastic dynamics characterized by uncertainties that are additive and control multiplicative. Stochastic models with the aforementioned characteristics have been used in computational neuroscience, biology, finance and aerospace systems and provide a more accurate representation of actuation than models with additive uncertainty. Previous literature has established the inadequacy of the linear HJB theory and instead rely on a non-linear Feynman-Kac lemma resulting in a second order forward-backward stochastic differential equations representation. However, the proposed solutions that use this representation suffer from compounding errors and computational complexity leading to lack of scalability. In this paper, we propose a deep learning based algorithm that leverages the second order Forward-Backward SDE representation and LSTM based recurrent neural networks to not only solve such Stochastic Optimal Control problems but also overcome the problems faced by previous approaches and scales well to high dimensional systems. The resulting control algorithm is tested on non-linear systems in robotics and biomechanics to demonstrate feasibility and out-performance against previous methods.

cs.LG↗