Substantial work indicates that the dynamics of neural networks (NNs) is closely related to their initialization of parameters. Inspired by the phase diagram for two-layer ReLU NNs with infinite width (Luo et al., 2021), we make a step towards drawing a phase diagram for three-layer ReLU NNs with infinite width. First, we derive a normalized gradient flow for three-layer ReLU NNs and obtain two key independent quantities to distinguish different dynamical regimes for common initialization methods. With carefully designed experiments and a large computation cost, for both synthetic datasets and real datasets, we find that the dynamics of each layer also could be divided into a linear regime and a condensed regime, separated by a critical regime. The criteria is the relative change of input weights (the input weight of a hidden neuron consists of the weight from its input layer to the hidden neuron and its bias term) as the width approaches infinity during the training, which tends to $0$, $+\infty$ and $O(1)$, respectively. In addition, we also demonstrate that different layers can lie in different dynamical regimes in a training process within a deep NN. In the condensed regime, we also observe the condensation of weights in isolated orientations with low complexity. Through experiments under three-layer condition, our phase diagram suggests a complicated dynamical regimes consisting of three possible regimes, together with their mixture, for deep NNs and provides a guidance for studying deep NNs in different initialization regimes, which reveals the possibility of completely different dynamics emerging within a deep NN for its different layers.

Substantial work indicates that the dynamics of neural networks (NNs) is closely related to their initialization of parameters. Inspired by the phase diagram for two-layer ReLU NNs with infinite width ( Luo et al., 2021), we make a step towards drawing a phase diagram for three-layer ReLU NNs with infinite width. First, we derive a normalized gradient flow for three-layer ReLU NNs and obtain two key independent quantities to distinguish different dynamical regimes for common initialization methods. With carefully designed experiments and a large computation cost, for both synthetic datasets and real datasets, we find that the dynamics of each layer also could be divided into a linear regime and a condensed regime, separated by a critical regime. The criteria is the relative change of input weights (the input weight of a hidden neuron consists of the weight from its input layer to the hidden neuron and its bias term) as the width approaches infinity during the training, which tends to 0, + and O (1), respectively. In addition, we also demonstrate that different layers can lie in different dynamical regimes in a training process within a deep NN. In the condensed regime, we also observe the condensation of weights in isolated orientations with low complexity. Through experiments under three-layer condition, our phase diagram suggests a complicated dynamical regimes consisting of three possible regimes, together with their mixture, for deep NNs and provides a guidance for studying deep NNs in different initialization regimes, which reveals the possibility of completely different dynamics emerging within a deep NN for its different layers.

Dynamical systems in the life sciences are often composed of complex mixtures of overlapping behavioral regimes. Cellular subpopulations may shift from cycling to equilibrium dynamics or branch towards different developmental fates. The transitions between these regimes can appear noisy and irregular, posing a serious challenge to traditional, flow-based modeling techniques which assume locally smooth dynamics. To address this challenge, we propose MODE (Mixture Of Dynamical Experts), a graphical modeling framework whose neural gating mechanism decomposes complex dynamics into sparse, interpretable components, enabling both the unsupervised discovery of behavioral regimes and accurate long-term forecasting across regime transitions. Crucially, because agents in our framework can jump to different governing laws, MODE is especially tailored to the aforementioned noisy transitions. We evaluate our method on a battery of synthetic and real datasets from computational biology. First, we systematically benchmark MODE on an unsupervised classification task using synthetic dynamical snapshot data, including in noisy, few-sample settings. Next, we show how MODE succeeds on challenging forecasting tasks which simulate key cycling and branching processes in cell biology. Finally, we deploy our method on human, single-cell RNA sequencing data and show that it can not only distinguish proliferation from differentiation dynamics but also predict when cells will commit to their ultimate fate, a key outstanding challenge in computational biology.

Substantial work indicates that the dynamics of neural networks (NNs) is closely related to their initialization of parameters. Inspired by the phase diagram for two-layer ReLU NNs with infinite width ( Luo et al., 2021), we make a step towards drawing a phase diagram for three-layer ReLU NNs with infinite width. First, we derive a normalized gradient flow for three-layer ReLU NNs and obtain two key independent quantities to distinguish different dynamical regimes for common initialization methods. With carefully designed experiments and a large computation cost, for both synthetic datasets and real datasets, we find that the dynamics of each layer also could be divided into a linear regime and a condensed regime, separated by a critical regime. The criteria is the relative change of input weights (the input weight of a hidden neuron consists of the weight from its input layer to the hidden neuron and its bias term) as the width approaches infinity during the training, which tends to 0, + and O (1), respectively. In addition, we also demonstrate that different layers can lie in different dynamical regimes in a training process within a deep NN. In the condensed regime, we also observe the condensation of weights in isolated orientations with low complexity. Through experiments under three-layer condition, our phase diagram suggests a complicated dynamical regimes consisting of three possible regimes, together with their mixture, for deep NNs and provides a guidance for studying deep NNs in different initialization regimes, which reveals the possibility of completely different dynamics emerging within a deep NN for its different layers.

The inductive bias and generalization properties of large machine learning models are -- to a substantial extent -- a byproduct of the optimization algorithm used for training. Among others, the scale of the random initialization, the learning rate, and early stopping all have crucial impact on the quality of the model learnt by stochastic gradient descent or related algorithms. In order to understand these phenomena, we study the training dynamics of large two-layer neural networks. We use a well-established technique from non-equilibrium statistical physics (dynamical mean field theory) to obtain an asymptotic high-dimensional characterization of this dynamics. This characterization applies to a Gaussian approximation of the hidden neurons non-linearity, and empirically captures well the behavior of actual neural network models. Our analysis uncovers several interesting new phenomena in the training dynamics: $(i)$ The emergence of a slow time scale associated with the growth in Gaussian/Rademacher complexity; $(ii)$ As a consequence, algorithmic inductive bias towards small complexity, but only if the initialization has small enough complexity; $(iii)$ A separation of time scales between feature learning and overfitting; $(iv)$ A non-monotone behavior of the test error and, correspondingly, a `feature unlearning' phase at large times.

Substantial work indicates that the dynamics of neural networks (NNs) is closely related to their initialization of parameters. Inspired by the phase diagram for two-layer ReLU NNs with infinite width (Luo et al., 2021), we make a step towards drawing a phase diagram for three-layer ReLU NNs with infinite width. First, we derive a normalized gradient flow for three-layer ReLU NNs and obtain two key independent quantities to distinguish different dynamical regimes for common initialization methods. With carefully designed experiments and a large computation cost, for both synthetic datasets and real datasets, we find that the dynamics of each layer also could be divided into a linear regime and a condensed regime, separated by a critical regime. The criteria is the relative change of input weights (the input weight of a hidden neuron consists of the weight from its input layer to the hidden neuron and its bias term) as the width approaches infinity during the training, which tends to 0, \infty and O(1), respectively.

Surrogate modeling of non-linear oscillator networks remains challenging due to discrepancies between simplified analytical models and real-world complexity. To bridge this gap, we investigate hybrid reservoir computing, combining reservoir computing with "expert" analytical models. Simulating the absence of an exact model, we first test the surrogate models with parameter errors in their expert model. Second, we assess their performance when their expert model lacks key non-linear coupling terms present in an extended ground-truth model. We focus on short-term forecasting across diverse dynamical regimes, evaluating the use of these surrogates for control applications. We show that hybrid reservoir computers generally outperform standard reservoir computers and exhibit greater robustness to parameter tuning. Notably, unlike standard reservoir computers, the performance of the hybrid does not degrade when crossing an observed spectral radius threshold. Furthermore, there is good performance for dynamical regimes not accessible to the expert model, demonstrating the contribution of the reservoir.

In science, we are often interested in obtaining a generative model of the underlying system dynamics from observed time series. While powerful methods for dynamical systems reconstruction (DSR) exist when data come from a single domain, how to best integrate data from multiple dynamical regimes and leverage it for generalization is still an open question. This becomes particularly important when individual time series are short, and group-level information may help to fill in for gaps in single-domain data. At the same time, averaging is not an option in DSR, as it will wipe out crucial dynamical properties (e.g., limit cycles in one domain vs. chaos in another). Hence, a framework is needed that enables to efficiently harvest group-level (multi-domain) information while retaining all single-domain dynamical characteristics. Here we provide such a hierarchical approach and showcase it on popular DSR benchmarks, as well as on neuroscientific and medical time series. In addition to faithful reconstruction of all individual dynamical regimes, our unsupervised methodology discovers common low-dimensional feature spaces in which datasets with similar dynamics cluster. The features spanning these spaces were further dynamically highly interpretable, surprisingly in often linear relation to control parameters that govern the dynamics of the underlying system. Finally, we illustrate transfer learning and generalization to new parameter regimes.

Complex ocean systems such as the Antarctic Circumpolar Current play key roles in the climate, and current models predict shifts in their strength and area under climate change. However, the physical processes underlying these changes are not well understood, in part due to the difficulty of characterizing and tracking changes in ocean physics in complex models. Using the Antarctic Circumpolar Current as a case study, we extend the method Tracking global Heating with Ocean Regimes (THOR) to a mesoscale eddy permitting climate model and identify regions of the ocean characterized by similar physics, called dynamical regimes, using readily accessible fields from climate models. To this end, we cluster grid cells into dynamical regimes and train an ensemble of neural networks, allowing uncertainty quantification, to predict these regimes and track them under climate change. Finally, we leverage this new knowledge to elucidate the dynamical drivers of the identified regime shifts as noted by the neural network using the 'explainability' methods SHAP and Layer-wise Relevance Propagation. A region undergoing a profound shift is where the Antarctic Circumpolar Current intersects the Pacific-Antarctic Ridge, an area important for carbon draw-down and fisheries. In this region, THOR specifically reveals a shift in dynamical regime under climate change driven by changes in wind stress and interactions with bathymetry. Using this knowledge to guide further exploration, we find that as the Antarctic Circumpolar Current shifts north under intensifying wind stress, the dominant dynamical role of bathymetry weakens and the flow intensifies.

Reservoir Computing is an emerging machine learning framework which is a versatile option for utilising physical systems for computation. In this paper, we demonstrate how a single node reservoir, made of a simple electronic circuit, can be employed for computation and explore the available options to improve the computational capability of the physical reservoirs. We build a reservoir computing system using a memristive chaotic oscillator as the reservoir. We choose two of the available hyperparameters to find the optimal working regime for the reservoir, resulting in two reservoir versions. We compare the performance of both the reservoirs in a set of three non-temporal tasks: approximating two non-chaotic polynomials and a chaotic trajectory of the Lorenz time series. We also demonstrate how the dynamics of the physical system plays a direct role in the reservoir's hyperparameters and hence in the reservoir's prediction ability.