Pantelis R. Vlachas

Pantelis R. Vlachas, Petros Koumoutsakos

Recurrent Neural Networks (RNNs) have become an integral part of modeling and forecasting frameworks in areas like natural language processing and high-dimensional dynamical systems such as turbulent fluid flows. To improve the accuracy of predictions, RNNs are trained using the Backpropagation Through Time (BPTT) method to minimize prediction loss. During testing, RNNs are often used in autoregressive scenarios where the output of the network is fed back into the input. However, this can lead to the exposure bias effect, as the network was trained to receive ground-truth data instead of its own predictions. This mismatch between training and testing is compounded when the state distributions are different, and the train and test losses are measured. To address this, previous studies have proposed solutions for language processing networks with probabilistic predictions. Building on these advances, we propose the Scheduled Autoregressive BPTT (BPTT-SA) algorithm for predicting complex systems. Our results show that BPTT-SA effectively reduces iterative error propagation in Convolutional RNNs and Convolutional Autoencoder RNNs, and demonstrate its capabilities in long-term prediction of high-dimensional fluid flows.

Ivica Kičić, Pantelis R. Vlachas, Georgios Arampatzis et al.

Predictive simulations are essential for applications ranging from weather forecasting to material design. The veracity of these simulations hinges on their capacity to capture the effective system dynamics. Massively parallel simulations predict the systems dynamics by resolving all spatiotemporal scales, often at a cost that prevents experimentation. On the other hand, reduced order models are fast but often limited by the linearization of the system dynamics and the adopted heuristic closures. We propose a novel systematic framework that bridges large scale simulations and reduced order models to extract and forecast adaptively the effective dynamics (AdaLED) of multiscale systems. AdaLED employs an autoencoder to identify reduced-order representations of the system dynamics and an ensemble of probabilistic recurrent neural networks (RNNs) as the latent time-stepper. The framework alternates between the computational solver and the surrogate, accelerating learned dynamics while leaving yet-to-be-learned dynamics regimes to the original solver. AdaLED continuously adapts the surrogate to the new dynamics through online training. The transitions between the surrogate and the computational solver are determined by monitoring the prediction accuracy and uncertainty of the surrogate. The effectiveness of AdaLED is demonstrated on three different systems - a Van der Pol oscillator, a 2D reaction-diffusion equation, and a 2D Navier-Stokes flow past a cylinder for varying Reynolds numbers (400 up to 1200), showcasing its ability to learn effective dynamics online, detect unseen dynamics regimes, and provide net speed-ups. To the best of our knowledge, AdaLED is the first framework that couples a surrogate model with a computational solver to achieve online adaptive learning of effective dynamics. It constitutes a potent tool for applications requiring many expensive simulations.

Pantelis R. Vlachas, Konstantinos Vlachas, Eleni Chatzi

Dynamical systems play a key role in modeling, forecasting, and decision-making across a wide range of scientific domains. However, variations in system parameters, also referred to as parametric variability, can lead to drastically different model behavior and output, posing challenges for constructing models that generalize across parameter regimes. In this work, we introduce the Parametric Hypernetwork for Learning Interpolated Networks (PHLieNet), a framework that simultaneously learns: (a) a global mapping from the parameter space to a nonlinear embedding and (b) a mapping from the inferred embedding to the weights of a dynamics propagation network. The learned embedding serves as a latent representation that modulates a base network, termed the hypernetwork, enabling it to generate the weights of a target network responsible for forecasting the system's state evolution conditioned on the previous time history. By interpolating in the space of models rather than observations, PHLieNet facilitates smooth transitions across parameterized system behaviors, enabling a unified model that captures the dynamic behavior across a broad range of system parameterizations. The performance of the proposed technique is validated in a series of dynamical systems with respect to its ability to extrapolate in time and interpolate and extrapolate in the parameter space, i.e., generalize to dynamics that were unseen during training. In all cases, our approach outperforms or matches state-of-the-art baselines in both short-term forecast accuracy and in capturing long-term dynamical features, such as attractor statistics.

Junaid Farooq, Danish Rafiq, Pantelis R. Vlachas et al.

Forecasting complex system dynamics, particularly for long-term predictions, is persistently hindered by error accumulation and computational burdens. This study presents RefreshNet, a multiscale framework developed to overcome these challenges, delivering an unprecedented balance between computational efficiency and predictive accuracy. RefreshNet incorporates convolutional autoencoders to identify a reduced order latent space capturing essential features of the dynamics, and strategically employs multiple recurrent neural network (RNN) blocks operating at varying temporal resolutions within the latent space, thus allowing the capture of latent dynamics at multiple temporal scales. The unique "refreshing" mechanism in RefreshNet allows coarser blocks to reset inputs of finer blocks, effectively controlling and alleviating error accumulation. This design demonstrates superiority over existing techniques regarding computational efficiency and predictive accuracy, especially in long-term forecasting. The framework is validated using three benchmark applications: the FitzHugh-Nagumo system, the Reaction-Diffusion equation, and Kuramoto-Sivashinsky dynamics. RefreshNet significantly outperforms state-of-the-art methods in long-term forecasting accuracy and speed, marking a significant advancement in modeling complex systems and opening new avenues in understanding and predicting their behavior.

Pantelis R. Vlachas, Julija Zavadlav, Matej Praprotnik et al.

Simulations are vital for understanding and predicting the evolution of complex molecular systems. However, despite advances in algorithms and special purpose hardware, accessing the timescales necessary to capture the structural evolution of bio-molecules remains a daunting task. In this work we present a novel framework to advance simulation timescales by up to three orders of magnitude, by learning the effective dynamics (LED) of molecular systems. LED augments the equation-free methodology by employing a probabilistic mapping between coarse and fine scales using mixture density network (MDN) autoencoders and evolves the non-Markovian latent dynamics using long short-term memory MDNs. We demonstrate the effectiveness of LED in the Müeller-Brown potential, the Trp Cage protein, and the alanine dipeptide. LED identifies explainable reduced-order representations and can generate, at any instant, the respective all-atom molecular trajectories. We believe that the proposed framework provides a dramatic increase to simulation capabilities and opens new horizons for the effective modeling of complex molecular systems.

Francesco Varoli, Guido Novati, Pantelis R. Vlachas et al.

We propose Improved Memories Learning (IMeL), a novel algorithm that turns reinforcement learning (RL) into a supervised learning (SL) problem and delimits the role of neural networks (NN) to interpolation. IMeL consists of two components. The first is a reservoir of experiences. Each experience is updated based on a non-parametric procedural improvement of the policy, computed as a bounded one-sample Monte Carlo estimate. The second is a NN regressor, which receives as input improved experiences from the reservoir (context points) and computes the policy by interpolation. The NN learns to measure the similarity between states in order to compute long-term forecasts by averaging experiences, rather than by encoding the problem structure in the NN parameters. We present preliminary results and propose IMeL as a baseline method for assessing the merits of more complex models and inductive biases.

Pantelis R. Vlachas, Georgios Arampatzis, Caroline Uhler et al.

Predictive simulations of complex systems are essential for applications ranging from weather forecasting to drug design. The veracity of these predictions hinges on their capacity to capture the effective system dynamics. Massively parallel simulations predict the system dynamics by resolving all spatiotemporal scales, often at a cost that prevents experimentation while their findings may not allow for generalisation. On the other hand reduced order models are fast but limited by the frequently adopted linearization of the system dynamics and/or the utilization of heuristic closures. Here we present a novel systematic framework that bridges large scale simulations and reduced order models to Learn the Effective Dynamics (LED) of diverse complex systems. The framework forms algorithmic alloys between non-linear machine learning algorithms and the Equation-Free approach for modeling complex systems. LED deploys autoencoders to formulate a mapping between fine and coarse-grained representations and evolves the latent space dynamics using recurrent neural networks. The algorithm is validated on benchmark problems and we find that it outperforms state of the art reduced order models in terms of predictability and large scale simulations in terms of cost. LED is applicable to systems ranging from chemistry to fluid mechanics and reduces the computational effort by up to two orders of magnitude while maintaining the prediction accuracy of the full system dynamics. We argue that LED provides a novel potent modality for the accurate prediction of complex systems.

Pantelis R. Vlachas, Jaideep Pathak, Brian R. Hunt et al.

We examine the efficiency of Recurrent Neural Networks in forecasting the spatiotemporal dynamics of high dimensional and reduced order complex systems using Reservoir Computing (RC) and Backpropagation through time (BPTT) for gated network architectures. We highlight advantages and limitations of each method and discuss their implementation for parallel computing architectures. We quantify the relative prediction accuracy of these algorithms for the longterm forecasting of chaotic systems using as benchmarks the Lorenz-96 and the Kuramoto-Sivashinsky (KS) equations. We find that, when the full state dynamics are available for training, RC outperforms BPTT approaches in terms of predictive performance and in capturing of the long-term statistics, while at the same time requiring much less training time. However, in the case of reduced order data, large scale RC models can be unstable and more likely than the BPTT algorithms to diverge. In contrast, RNNs trained via BPTT show superior forecasting abilities and capture well the dynamics of reduced order systems. Furthermore, the present study quantifies for the first time the Lyapunov Spectrum of the KS equation with BPTT, achieving similar accuracy as RC. This study establishes that RNNs are a potent computational framework for the learning and forecasting of complex spatiotemporal systems.

Pantelis R. Vlachas, Wonmin Byeon, Zhong Y. Wan et al.

We introduce a data-driven forecasting method for high-dimensional chaotic systems using long short-term memory (LSTM) recurrent neural networks. The proposed LSTM neural networks perform inference of high-dimensional dynamical systems in their reduced order space and are shown to be an effective set of nonlinear approximators of their attractor. We demonstrate the forecasting performance of the LSTM and compare it with Gaussian processes (GPs) in time series obtained from the Lorenz 96 system, the Kuramoto-Sivashinsky equation and a prototype climate model. The LSTM networks outperform the GPs in short-term forecasting accuracy in all applications considered. A hybrid architecture, extending the LSTM with a mean stochastic model (MSM-LSTM), is proposed to ensure convergence to the invariant measure. This novel hybrid method is fully data-driven and extends the forecasting capabilities of LSTM networks.