Linking Microscopic and Macroscopic Models for Evolution: Markov Chain Network Training and Conservation Law Approximations

We apply the theory of learning to physically renormalizable systems in an attempt to develop a theory of biological evolution, including the origin of life, as multilevel learning. We formulate seven fundamental principles of evolution that appear to be necessary and sufficient to render a universe observable and show that they entail the major features of biological evolution, including replication and natural selection. These principles also follow naturally from the theor...

Complex high dimensional stochastic dynamic systems arise in many applications in the natural sciences and especially biology. However, while these systems are difficult to describe analytically, "snapshot" measurements that sample the output of the system are often available. In order to model the dynamics of such systems given snapshot data, or local transitions, we present a deep neural network framework we call Dynamics Modeling Network or DyMoN. DyMoN is a neural network...

The process of training an artificial neural network involves iteratively adapting its parameters so as to minimize the error of the network's prediction, when confronted with a learning task. This iterative change can be naturally interpreted as a trajectory in network space -- a time series of networks -- and thus the training algorithm (e.g. gradient descent optimization of a suitable loss function) can be interpreted as a dynamical system in graph space. In order to illus...

Can a micron sized sack of interacting molecules understand, and adapt to a constantly-fluctuating environment? Cellular life provides an existence proof in the affirmative, but the principles that allow for life's existence are far from being proven. One challenge in engineering and understanding biochemical computation is the intrinsic noise due to chemical fluctuations. In this paper, we draw insights from machine learning theory, chemical reaction network theory, and stat...

In the brain, learning signals change over time and synaptic location, and are applied based on the learning history at the synapse, in the complex process of neuromodulation. Learning in artificial neural networks, on the other hand, is shaped by hyper-parameters set before learning starts, which remain static throughout learning, and which are uniform for the entire network. In this work, we propose a method of deep artificial neuromodulation which applies the concepts of b...

We introduce a methodology for seeking conservation laws within a Hamiltonian dynamical system, which we term ``neural deflation''. Inspired by deflation methods for steady states of dynamical systems, we propose to {iteratively} train a number of neural networks to minimize a regularized loss function accounting for the necessity of conserved quantities to be {\it in involution} and enforcing functional independence thereof consistently in the infinite-sample limit. The meth...

We review several of the most widely used techniques for training recurrent neural networks to approximate dynamical systems, then describe a novel algorithm for this task. The algorithm is based on an earlier theoretical result that guarantees the quality of the network approximation. We show that a feedforward neural network can be trained on the vector field representation of a given dynamical system using backpropagation, then recast, using matrix manipulations, as a recu...

We present a stochastic first-order optimization method specialized for deep neural networks (DNNs), ECCO-DNN. This method models the optimization variable trajectory as a dynamical system and develops a discretization algorithm that adaptively selects step sizes based on the trajectory's shape. This provides two key insights: designing the dynamical system for fast continuous-time convergence and developing a time-stepping algorithm to adaptively select step sizes based on p...

Understanding the dynamics of neural network parameters during training is one of the key challenges in building a theoretical foundation for deep learning. A central obstacle is that the motion of a network in high-dimensional parameter space undergoes discrete finite steps along complex stochastic gradients derived from real-world datasets. We circumvent this obstacle through a unifying theoretical framework based on intrinsic symmetries embedded in a network's architecture...

The design of deep neural networks remains somewhat of an art rather than precise science. By tentatively adopting ergodic theory considerations on top of viewing the network as the time evolution of a dynamical system, with each layer corresponding to a temporal instance, we show that some rules of thumb, which might otherwise appear mysterious, can be attributed heuristics.