arXiv subjects

Yuhai Tu

Publications and source records attributed to Yuhai Tu.

At least 37 records · Page 2Linked to original sources

arXiv2022

Free energy dissipation enhances spatial accuracy and robustness of Turing pattern in small reaction-diffusion systems

Accurate and robust spatial orders are ubiquitous in living systems. In 1952, Alan Turing proposed an elegant mechanism for pattern formation based on spontaneous breaking of the spatial translational symmetry in the underlying reaction-diffusion system. Much is understood about dynamics and structure of Turing patterns. However, little is known about the energetic cost of Turing pattern. Here, we study nonequilibrium thermodynamics of a small spatially extended biochemical reaction-diffusion system by using analytical and numerical methods. We find that the onset of Turing pattern requires a minimum energy dissipation to drive the nonequilibrium chemical reactions. Above onset, only a small fraction of the total energy expenditure is used to overcome diffusion for maintaining the spatial pattern. We show that the positioning error decreases as energy dissipation increases following the same tradeoff relationship between timing error and energy cost in biochemical oscillatory systems. In a finite system, we find that a specific Turing pattern exists only within a finite range of total molecule number, and energy dissipation broadens the range, which enhances the robustness of the Turing pattern against molecule number fluctuations in living cells. These results are verified in a realistic model of the Muk system underlying DNA segregation in E. coli, and testable predictions are made for the dependence of the accuracy and robustness of the spatial pattern on the ATP/ADP ratio. In general, the theoretical framework developed here can be applied to study nonequilibrium thermodynamics of spatially extended biochemical systems.

cond-mat.stat-mech↗

arXiv2022

Stochastic gradient descent introduces an effective landscape-dependent regularization favoring flat solutions

Generalization is one of the most important problems in deep learning (DL). In the overparameterized regime in neural networks, there exist many low-loss solutions that fit the training data equally well. The key question is which solution is more generalizable. Empirical studies showed a strong correlation between flatness of the loss landscape at a solution and its generalizability, and stochastic gradient descent (SGD) is crucial in finding the flat solutions. To understand how SGD drives the learning system to flat solutions, we construct a simple model whose loss landscape has a continuous set of degenerate (or near degenerate) minima. By solving the Fokker-Planck equation of the underlying stochastic learning dynamics, we show that due to its strong anisotropy the SGD noise introduces an additional effective loss term that decreases with flatness and has an overall strength that increases with the learning rate and batch-to-batch variation. We find that the additional landscape-dependent SGD-loss breaks the degeneracy and serves as an effective regularization for finding flat solutions. Furthermore, a stronger SGD noise shortens the convergence time to the flat solutions. However, we identify an upper bound for the SGD noise beyond which the system fails to converge. Our results not only elucidate the role of SGD for generalization they may also have important implications for hyperparameter selection for learning efficiently without divergence.

cond-mat.dis-nn↗

arXiv2022

State-space renormalization group theory of nonequilibrium reaction networks: Exact solutions for hypercubic lattices in arbitrary dimensions

Nonequilibrium reaction networks (NRNs) underlie most biological functions. Despite their diverse dynamic properties, NRNs share the signature characteristics of persistent probability fluxes and continuous energy dissipation even in the steady state. Dynamics of NRNs can be described at different coarse-grained levels. Our previous work showed that the apparent energy dissipation rate at a coarse-grained level follows an inverse power law dependence on the scale of coarse-graining. The scaling exponent is determined by the network structure and correlation of stationary probability fluxes. However, it remains unclear whether and how the (renormalized) flux correlation varies with coarse-graining. Following Kadanoff's real space renormalization group (RG) approach for critical phenomena, we address this question by developing a State-Space Renormalization Group (SSRG) theory for NRNs, which leads to an iterative RG equation for the flux correlation function. In square and hypercubic lattices, we solve the RG equation exactly and find two types of fixed point solutions: a family of nontrivial fixed points where the correlation exhibits power-law decay and a trivial fixed point where the correlation vanishes beyond the nearest neighbors. The power-law fixed point is stable if and only if the power exponent is less than the lattice dimension $n$. Consequently, the correlation function converges to the power-law fixed point only when the correlation in the fine-grained network decays slower than $r^{-n}$ and to the trivial fixed point otherwise. If the flux correlation in the fine-grained network contains multiple stable solutions with different exponents, the RG iteration dynamics select the fixed point solution with the smallest exponent. We also discuss a possible connection between the RG flows of flux correlation with those of the Kosterlitz-Thouless transition.

cond-mat.stat-mech↗

arXiv2022

Modeling bacterial flagellar motor with new structure information: Rotational dynamics of two interacting protein nano-rings

In this article, we develop a mathematical model for the rotary bacterial flagellar motor (BFM) based on the recently discovered structure of the stator complex (MotA$_5$MotB$_2$). The structure suggested that the stator also rotates. The BFM is modeled as two rotating nano-rings that interact with each other. Specifically, translocation of protons through the stator complex drives rotation of the MotA pentamer ring, which in turn drives rotation of the FliG ring in the rotor via interactions between the MotA ring of the stator and the FliG ring of the rotor. Preliminary results from the structure-informed model are consistent with the observed torque-speed relation. More importantly, the model predicts distinctive rotor and stator dynamics and their load dependence, which may be tested by future experiments. Possible approaches to verify and improve the model to further understanding of the molecular mechanism for torque generation in BFM are also discussed.

cond-mat.soft↗

arXiv2021

Loss Landscape Dependent Self-Adjusting Learning Rates in Decentralized Stochastic Gradient Descent

Distributed Deep Learning (DDL) is essential for large-scale Deep Learning (DL) training. Synchronous Stochastic Gradient Descent (SSGD) 1 is the de facto DDL optimization method. Using a sufficiently large batch size is critical to achieving DDL runtime speedup. In a large batch setting, the learning rate must be increased to compensate for the reduced number of parameter updates. However, a large learning rate may harm convergence in SSGD and training could easily diverge. Recently, Decentralized Parallel SGD (DPSGD) has been proposed to improve distributed training speed. In this paper, we find that DPSGD not only has a system-wise run-time benefit but also a significant convergence benefit over SSGD in the large batch setting. Based on a detailed analysis of the DPSGD learning dynamics, we find that DPSGD introduces additional landscape-dependent noise that automatically adjusts the effective learning rate to improve convergence. In addition, we theoretically show that this noise smoothes the loss landscape, hence allowing a larger learning rate. We conduct extensive studies over 18 state-of-the-art DL models/tasks and demonstrate that DPSGD often converges in cases where SSGD diverges for large learning rates in the large batch setting. Our findings are consistent across two different application domains: Computer Vision (CIFAR10 and ImageNet-1K) and Automatic Speech Recognition (SWB300 and SWB2000), and two different types of neural network models: Convolutional Neural Networks and Long Short-Term Memory Recurrent Neural Networks.

cs.LG↗

arXiv2021

Multiplicative noise underlies Taylor's law in protein concentration fluctuations in single cells

Protein concentration in a living cell fluctuates over time due to noise in growth and division processes. From extensive single-cell experiments by using E. coli strains with different promoter strength (over two orders of magnitude) and under different nutrient conditions, we found that the variance of protein concentration fluctuations follows a robust square power-law dependence on its mean, which belongs to a general phenomenon called Taylor's law. To understand the mechanistic origin of this observation, we use a minimal mechanistic model to describe the stochastic growth and division processes in a single cell with a feedback mechanism for regulating cell division. The model reproduces the observed Taylor's law. The predicted protein concentration distributions agree quantitatively with those measured in experiments for different nutrient conditions and a wide range of promoter strength. By using a mean-field approximation, we derived a single Langevin equation for protein concentration with multiplicative noise, which can be solved exactly to prove the square Taylor's law and to obtain an analytical solution for the protein concentration distribution function that agrees with experiments. Analysis of experiments by using our model showed that noise in production rates dominates over noise from cell division in contributing to protein concentration fluctuations. In general, a multiplicative noise in the underlying stochastic dynamics may be responsible for Taylor's law in other systems.

physics.bio-ph↗

arXiv2021

Phases of learning dynamics in artificial neural networks: with or without mislabeled data

Despite tremendous success of deep neural network in machine learning, the underlying reason for its superior learning capability remains unclear. Here, we present a framework based on statistical physics to study dynamics of stochastic gradient descent (SGD) that drives learning in neural networks. By using the minibatch gradient ensemble, we construct order parameters to characterize dynamics of weight updates in SGD. Without mislabeled data, we find that the SGD learning dynamics transitions from a fast learning phase to a slow exploration phase, which is associated with large changes in order parameters that characterize the alignment of SGD gradients and their mean amplitude. In the case with randomly mislabeled samples, SGD learning dynamics falls into four distinct phases. The system first finds solutions for the correctly labeled samples in phase I, it then wanders around these solutions in phase II until it finds a direction to learn the mislabeled samples during phase III, after which it finds solutions that satisfy all training samples during phase IV. Correspondingly, the test error decreases during phase I and remains low during phase II; however, it increases during phase III and reaches a high plateau during phase IV. The transitions between different phases can be understood by changes of order parameters that characterize the alignment of mean gradients for the correctly and incorrectly labeled samples and their (relative) strength during learning. We find that individual sample losses for the two datasets are most separated during phase II, which leads to a cleaning process to eliminate mislabeled samples for improving generalization.

cs.LG↗

arXiv2020

Scaling of Energy Dissipation in Nonequilibrium Reaction Networks

The energy dissipation rate in a nonequilibirum reaction system can be determined by the reaction rates in the underlying reaction network. By developing a coarse-graining process in state space and a corresponding renormalization procedure for reaction rates, we find that energy dissipation rate has an inverse power-law dependence on the number of microscopic states in a coarse-grained state. The dissipation scaling law requires self-similarity of the underlying network, and the scaling exponent depends on the network structure and the flux correlation. Implications of this inverse dissipation scaling law for active flow systems such as microtubule-kinesin mixture are discussed.

cond-mat.stat-mech↗

arXiv2020

Deciphering gene regulation from gene expression dynamics using deep neural network

Complex biological functions are carried out by the interaction of genes and proteins. Uncovering the gene regulation network behind a function is one of the central themes in biology. Typically, it involves extensive experiments of genetics, biochemistry and molecular biology. In this paper, we show that much of the inference task can be accomplished by a deep neural network (DNN), a form of machine learning or artificial intelligence. Specifically, the DNN learns from the dynamics of the gene expression. The learnt DNN behaves like an accurate simulator of the system, on which one can perform in-silico experiments to reveal the underlying gene network. We demonstrate the method with two examples: biochemical adaptation and the gap-gene patterning in fruit fly embryogenesis. In the first example, the DNN can successfully find the two basic network motifs for adaptation - the negative feedback and the incoherent feed-forward. In the second and much more complex example, the DNN can accurately predict behaviors of essentially all the mutants. Furthermore, the regulation network it uncovers is strikingly similar to the one inferred from experiments. In doing so, we develop methods for deciphering the gene regulation network hidden in the DNN "black box". Our interpretable DNN approach should have broad applications in genotype-phenotype mapping.

q-bio.QM↗

arXiv2020

Nonequilibrium thermodynamics of coupled molecular oscillators: The energy cost and optimal design for synchronization

A model of coupled molecular oscillators is proposed to study nonequilibrium thermodynamics of synchronization. We find that synchronization of nonequilibrium oscillators costs energy even when the oscillator-oscillator coupling is conservative. By solving the steady state of the many-body system analytically, we show that the system goes through a nonequilibrium phase transition driven by energy dissipation, and the critical energy dissipation depends on both the frequency and strength of the exchange reactions. Moreover, our study reveals the optimal design for achieving maximum synchronization with a fixed energy budget. We apply our general theory to the Kai system in Cyanobacteria circadian clock and predict a relationship between the KaiC ATPase activity and synchronization of the KaiC hexamers. The theoretical framework can be extended to study thermodynamics of collective behaviors in other extended nonequilibrium active systems.

physics.bio-ph↗

arXiv2020

How neural networks find generalizable solutions: Self-tuned annealing in deep learning

Despite the tremendous success of Stochastic Gradient Descent (SGD) algorithm in deep learning, little is known about how SGD finds generalizable solutions in the high-dimensional weight space. By analyzing the learning dynamics and loss function landscape, we discover a robust inverse relation between the weight variance and the landscape flatness (inverse of curvature) for all SGD-based learning algorithms. To explain the inverse variance-flatness relation, we develop a random landscape theory, which shows that the SGD noise strength (effective temperature) depends inversely on the landscape flatness. Our study indicates that SGD attains a self-tuned landscape-dependent annealing strategy to find generalizable solutions at the flat minima of the landscape. Finally, we demonstrate how these new theoretical insights lead to more efficient algorithms, e.g., for avoiding catastrophic forgetting.

physics.data-an↗

arXiv2019

Error-speed correlations in biopolymer synthesis

Synthesis of biopolymers such as DNA, RNA, and proteins are biophysical processes aided by enzymes. Performance of these enzymes is usually characterized in terms of their average error rate and speed. However, because of thermal fluctuations in these single-molecule processes, both error and speed are inherently stochastic quantities. In this paper, we study fluctuations of error and speed in biopolymer synthesis and show that they are in general correlated. This means that, under equal conditions, polymers that are synthesized faster due to a fluctuation tend to have either better or worse errors than the average. The error-correction mechanism implemented by the enzyme determines which of the two cases holds. For example, discrimination in the forward reaction rates tends to grant smaller errors to polymers with faster synthesis. The opposite occurs for discrimination in monomer rejection rates. Our results provide an experimentally feasible way to identify error-correction mechanisms by measuring the error-speed correlations.

q-bio.SC↗

arXiv2019

Learning to Learn without Forgetting by Maximizing Transfer and Minimizing Interference

Lack of performance when it comes to continual learning over non-stationary distributions of data remains a major challenge in scaling neural network learning to more human realistic settings. In this work we propose a new conceptualization of the continual learning problem in terms of a temporally symmetric trade-off between transfer and interference that can be optimized by enforcing gradient alignment across examples. We then propose a new algorithm, Meta-Experience Replay (MER), that directly exploits this view by combining experience replay with optimization based meta-learning. This method learns parameters that make interference based on future gradients less likely and transfer based on future gradients more likely. We conduct experiments across continual lifelong supervised learning benchmarks and non-stationary reinforcement learning environments demonstrating that our approach consistently outperforms recently proposed baselines for continual learning. Our experiments show that the gap between the performance of MER and baseline algorithms grows both as the environment gets more non-stationary and as the fraction of the total experiences stored gets smaller.

cs.LG↗

arXiv2019

Continual Learning with Self-Organizing Maps

Despite remarkable successes achieved by modern neural networks in a wide range of applications, these networks perform best in domain-specific stationary environments where they are trained only once on large-scale controlled data repositories. When exposed to non-stationary learning environments, current neural networks tend to forget what they had previously learned, a phenomena known as catastrophic forgetting. Most previous approaches to this problem rely on memory replay buffers which store samples from previously learned tasks, and use them to regularize the learning on new ones. This approach suffers from the important disadvantage of not scaling well to real-life problems in which the memory requirements become enormous. We propose a memoryless method that combines standard supervised neural networks with self-organizing maps to solve the continual learning problem. The role of the self-organizing map is to adaptively cluster the inputs into appropriate task contexts - without explicit labels - and allocate network resources accordingly. Thus, it selectively routes the inputs in accord with previous experience, ensuring that past learning is maintained and does not interfere with current learning. Out method is intuitive, memoryless, and performs on par with current state-of-the-art approaches on standard benchmarks.

cs.NE↗

arXiv2018

Hydrodynamic Theory of Flocking in the Presence of Quenched Disorder

The effect of quenched (frozen) orientational disorder on the collective motion of active particles is analyzed. We find that, as with annealed disorder (Langevin noise), active polar systems are far more robust against quenched disorder than their equilibrium counterparts. In particular, long ranged order (i.e., the existence of a non-zero average velocity $\langle {\bf v} \rangle$) persists in the presence of quenched disorder even in spatial dimensions $d=3$, while it is destroyed even by arbitrarily weak disorder in $d \le 4$ in equilibrium systems. Furthermore, in $d=2$, quasi-long-ranged order (i.e., spatial velocity correlations that decay as a power law with distance) occurs when quenched disorder is present, in contrast to the short-ranged order that is all that can survive in equilibrium. These predictions are borne out by simulations in both two and three dimensions.

cond-mat.stat-mech↗

arXiv2018

Swarming in the Dirt: Ordered Flocks with Quenched Disorder

The effect of quenched (frozen) disorder on the collective motion of active particles is analyzed. We find that active polar systems are far more robust against quenched disorder than equilibrium ferromagnets. Long ranged order (a non-zero average velocity $\langle{\bf v}\rangle$) persists in the presence of quenched disorder even in spatial dimensions $d=3$; in $d=2$, quasi-long-ranged order (i.e., spatial velocity correlations that decay as a power law with distance) occurs. In equilibrium systems, only quasi-long-ranged order in $d=3$ and short ranged order in $d=2$ are possible. Our theoretical predictions for two dimensions are borne out by simulations.

cond-mat.stat-mech↗

arXiv2018

Design principles and optimal performance for molecular motors under realistic constraints

The performance of a molecular motor, characterized by its power output and energy efficiency, is investigated in the motor design space spanned by the stepping rate function and the motor-track interaction potential. Analytic results and simulations show that a gating mechanism that restricts forward stepping in a narrow window in configuration space is needed for generating high power at physiologically relevant loads. By deriving general thermodynamics laws for nonequilibrium motors, we find that the maximum torque (force) at stall is less than its theoretical limit for any realistic motor-track interactions due to speed fluctuations. Our study reveals a tradeoff for the motor- track interaction: while a strong interaction generates a high power output for forward steps, it also leads to a higher probability of wasteful spontaneous back steps. Our analysis and simulations show that this tradeoff sets a fundamental limit to the maximum motor efficiency in the presence of spontaneous back steps, i.e., loose-coupling. Balancing this tradeoff leads to an optimal design of the motor-track interaction for achieving a maximum efficiency close to 1 for realistic motors that are not perfectly coupled with the energy source.Comparison with existing data and suggestions for future experiments are discussed.

cond-mat.soft↗

arXiv2017

Perfect and near perfect adaptation in a model of bacterial chemotaxis

The signaling apparatus mediating bacterial chemotaxis can adapt to a wide range of persistent external stimuli. In many cases, the bacterial activity returns to its pre-stimulus level exactly and this "perfect adaptability" is robust against variations in various chemotaxis protein concentrations. We model the bacterial chemotaxis signaling pathway, from ligand binding to CheY phosphorylation. By solving the steady-state equations of the model analytically, we derive a full set of conditions for the system to achieve perfect adaptation. The conditions related to the phosphorylation part of the pathway are discovered for the first time, while other conditions are generalization of the ones found in previous works. Sensitivity of the perfect adaptation is evaluated by perturbing these conditions. We find that, even in the absence of some of the perfect adaptation conditions, adaptation can be achieved with near perfect precision as a result of the separation of scales in both chemotaxis protein concentrations and reaction rates, or specific properties of the receptor distribution in different methylation states. Since near perfect adaptation can be found in much larger regions of the parameter space than that defined by the perfect adaptation conditions, their existence is essential to understand robustness in bacterial chemotaxis.

physics.bio-ph↗