Petros Koumoutsakos

Han Gao, Sebastian Kaltenbach, Petros Koumoutsakos

We introduce a generative learning framework to model high-dimensional parametric systems using gradient guidance and virtual observations. We consider systems described by Partial Differential Equations (PDEs) discretized with structured or unstructured grids. The framework integrates multi-level information to generate high fidelity time sequences of the system dynamics. We demonstrate the effectiveness and versatility of our framework with two case studies in incompressible, two dimensional, low Reynolds cylinder flow on an unstructured mesh and incompressible turbulent channel flow on a structured mesh, both parameterized by the Reynolds number. Our results illustrate the framework's robustness and ability to generate accurate flow sequences across various parameter settings, significantly reducing computational costs allowing for efficient forecasting and reconstruction of flow dynamics.

Emmanuel Menier, Sebastian Kaltenbach, Mouadh Yagoubi et al.

The modeling and simulation of high-dimensional multiscale systems is a critical challenge across all areas of science and engineering. It is broadly believed that even with today's computer advances resolving all spatiotemporal scales described by the governing equations remains a remote target. This realization has prompted intense efforts to develop model order reduction techniques. In recent years, techniques based on deep recurrent neural networks have produced promising results for the modeling and simulation of complex spatiotemporal systems and offer large flexibility in model development as they can incorporate experimental and computational data. However, neural networks lack interpretability, which limits their utility and generalizability across complex systems. Here we propose a novel framework of Interpretable Learning Effective Dynamics (iLED) that offers comparable accuracy to state-of-the-art recurrent neural network-based approaches while providing the added benefit of interpretability. The iLED framework is motivated by Mori-Zwanzig and Koopman operator theory, which justifies the choice of the specific architecture. We demonstrate the effectiveness of the proposed framework in simulations of three benchmark multiscale systems. Our results show that the iLED framework can generate accurate predictions and obtain interpretable dynamics, making it a promising approach for solving high-dimensional multiscale systems.

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.

Jonas Šukys, Ursula Rasthofer, Fabian Wermelinger et al.

We quantify uncertainties in the location and magnitude of extreme pressure spots revealed from large scale multi-phase flow simulations of cloud cavitation collapse. We examine clouds containing 500 cavities and quantify uncertainties related to their initial spatial arrangement. The resulting 2000-dimensional space is sampled using a non-intrusive and computationally efficient Multi-Level Monte Carlo (MLMC) methodology. We introduce novel optimal control variate coefficients to enhance the variance reduction in MLMC. The proposed optimal fidelity MLMC leads to more than two orders of magnitude speedup when compared to standard Monte Carlo methods. We identify large uncertainties in the location and magnitude of the peak pressure pulse and present its statistical correlations and joint probability density functions with the geometrical characteristics of the cloud. Characteristic properties of spatial cloud structure are identified as potential causes of significant uncertainties in exerted collapse pressures.

Pascal Weber, Daniel Wälchli, Mustafa Zeqiri et al.

We present the extension of the Remember and Forget for Experience Replay (ReF-ER) algorithm to Multi-Agent Reinforcement Learning (MARL). ReF-ER was shown to outperform state of the art algorithms for continuous control in problems ranging from the OpenAI Gym to complex fluid flows. In MARL, the dependencies between the agents are included in the state-value estimator and the environment dynamics are modeled via the importance weights used by ReF-ER. In collaborative environments, we find the best performance when the value is estimated using individual rewards and we ignore the effects of other actions on the transition map. We benchmark the performance of ReF-ER MARL on the Stanford Intelligent Systems Laboratory (SISL) environments. We find that employing a single feed-forward neural network for the policy and the value function in ReF-ER MARL, outperforms state of the art algorithms that rely on complex neural network architectures.

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.

Michal Balcerak, Tamaz Amiranashvili, Andreas Wagner et al.

Physical models in the form of partial differential equations serve as important priors for many under-constrained problems. One such application is tumor treatment planning, which relies on accurately estimating the spatial distribution of tumor cells within a patient's anatomy. While medical imaging can detect the bulk of a tumor, it cannot capture the full extent of its spread, as low-concentration tumor cells often remain undetectable, particularly in glioblastoma, the most common primary brain tumor. Machine learning approaches struggle to estimate the complete tumor cell distribution due to a lack of appropriate training data. Consequently, most existing methods rely on physics-based simulations to generate anatomically and physiologically plausible estimations. However, these approaches face challenges with complex and unknown initial conditions and are constrained by overly rigid physical models. In this work, we introduce a novel method that integrates data-driven and physics-based cost functions, akin to Physics-Informed Neural Networks (PINNs). However, our approach parametrizes the solution directly on a dynamic discrete mesh, allowing for the effective modeling of complex biomechanical behaviors. Specifically, we propose a unique discretization scheme that quantifies how well the learned spatiotemporal distributions of tumor and brain tissues adhere to their respective growth and elasticity equations. This quantification acts as a regularization term, offering greater flexibility and improved integration of patient data compared to existing models. We demonstrate enhanced coverage of tumor recurrence areas using real-world data from a patient cohort, highlighting the potential of our method to improve model-driven treatment planning for glioblastoma in clinical practice.

Georgios Arampatzis, Daniel Wälchli, Panagiotis Angelikopoulos et al.

We propose an algorithm for the efficient and robust sampling of the posterior probability distribution in Bayesian inference problems. The algorithm combines the local search capabilities of the Manifold Metropolis Adjusted Langevin transition kernels with the advantages of global exploration by a population based sampling algorithm, the Transitional Markov Chain Monte Carlo (TMCMC). The Langevin diffusion process is determined by either the Hessian or the Fisher Information of the target distribution with appropriate modifications for non positive definiteness. The present methods is shown to be superior over other population based algorithms, in sampling probability distributions for which gradients are available and is shown to handle otherwise unidentifiable models. We demonstrate the capabilities and advantages of the method in computing the posterior distribution of the parameters in a Pharmacodynamics model, for glioma growth and its drug induced inhibition, using clinical data.

Sebastian Kaltenbach, Phaedon-Stelios Koutsourelakis, Petros Koumoutsakos

Partial Differential Equations (PDEs) with high dimensionality are commonly encountered in computational physics and engineering. However, finding solutions for these PDEs can be computationally expensive, making model-order reduction crucial. We propose such a data-driven scheme that automates the identification of the time-scales involved and can produce stable predictions forward in time as well as under different initial conditions not included in the training data. To this end, we combine a non-linear autoencoder architecture with a time-continuous model for the latent dynamics in the complex space. It readily allows for the inclusion of sparse and irregularly sampled training data. The learned, latent dynamics are interpretable and reveal the different temporal scales involved. We show that this data-driven scheme can automatically learn the independent processes that decompose a system of linear ODEs along the eigenvectors of the system's matrix. Apart from this, we demonstrate the applicability of the proposed framework in a hidden Markov Model and the (discretized) Kuramoto-Shivashinsky (KS) equation. Additionally, we propose a probabilistic version, which captures predictive uncertainties and further improves upon the results of the deterministic framework.

Eneko Lazpita, Andrés Bell-Navas, Jesús Garicano-Mena et al.

Understanding intraventricular hemodynamics requires compact and physically interpretable representations of the underlying flow structures, as characteristic flow patterns are closely associated with cardiovascular conditions and can support early detection of cardiac deterioration. Conventional visualization of velocity or pressure fields, however, provides limited insight into the coherent mechanisms driving these dynamics. Reduced-order modeling techniques, like Proper Orthogonal Decomposition (POD) and Autoencoder (AE) architectures, offer powerful alternatives to extract dominant flow features from complex datasets. This study systematically compares POD with several AE variants (Linear, Nonlinear, Convolutional, and Variational) using left ventricular flow fields obtained from computational fluid dynamics simulations. We show that, for a suitably chosen latent dimension, AEs produce modes that become nearly orthogonal and qualitatively resemble POD modes that capture a given percentage of kinetic energy. As the number of latent modes increases, AE modes progressively lose orthogonality, leading to linear dependence, spatial redundancy, and the appearance of repeated modes with substantial high-frequency content. This degradation reduces interpretability and introduces noise-like components into AE-based reduced-order models, potentially complicating their integration with physics-based formulations or neural-network surrogates. The extent of interpretability loss varies across the AEs, with nonlinear, convolutional, and variational models exhibiting distinct behaviors in orthogonality preservation and feature localization. Overall, the results indicate that AEs can reproduce POD-like coherent structures under specific latent-space configurations, while highlighting the need for careful mode selection to ensure physically meaningful representations of cardiac flow dynamics.

Konstantinos Vogiatzoglou, Costas Papadimitriou, Vasilis Bontozoglou et al.

Forward propagation of input uncertainties in physics-based wildfire models is computationally prohibitive, limiting the use of high-fidelity simulators in risk assessment workflows. This work introduces a geometry-aligned bi-fidelity surrogate framework that addresses the convection-dominated nature of wildfire spread by mapping low- and high-fidelity solution snapshots onto a common reference domain prior to basis selection and reconstruction. Unlike conventional bi-fidelity schemes, which combine spatially shifted snapshots and thus suffer from oscillations and excess basis requirements near sharp fronts, the proposed mapping aligns the dominant front geometry through per-variable shift/stretch transforms in 1D and an activity indicator-based affine alignment in 2D, so that reduced bases compare physically corresponding structures rather than displaced ones. Building on the ADfiRe physics-based simulator, we demonstrate the method on 1D and 2D test cases in which low- and high-fidelity models differ in mesh resolution and physical completeness. Across both settings, the geometry-aligned surrogate reproduces full-field temperature and fuel composition with substantially lower error than its unmapped counterpart, eliminates Gibbs-type oscillations near steep gradients, and recovers high-fidelity probability density functions for key quantities of interest (e.g., maximum temperature, evaporated moisture, and burned area). After offline training, online predictions are roughly three orders of magnitude cheaper than direct high-fidelity evaluation, making the framework a practical building block for many-query uncertainty quantification once the offline cost is amortized over enough queries. We discuss the conditions under which the geometric alignment is most effective, its limitations for non-convex or topologically complex fronts, and the path toward validation against real data.

Aaron Miller, Sahil Kommalapati, Robert Moser et al.

Accurate and generalizable Reynolds-averaged Navier-Stokes (RANS) models for turbulent flows rely on effective closures. We introduce tensor-based, symmetry aware closures using equivariant neural networks (ENNs) and present an algorithm for enforcing algebraic contraction relations among tensor components. The modeling approach builds on the structure tensor framework introduced by Kassinos and Reynolds to learn closures in the rapid distortion theory setting. Experiments show that ENNs can effectively learn relationships involving high-order tensors, meeting or exceeding the performance of existing models in tasks such as predicting the rapid pressure-strain correlation. Our results show that ENNs provide a physically consistent alternative to classical tensor basis models, enabling end-to-end learning of unclosed terms in RANS and fast exploration of model dependencies.

Han Gao, Sebastian Kaltenbach, Petros Koumoutsakos

We introduce generative models for accelerating simulations of complex systems through learning and evolving their effective dynamics. In the proposed Generative Learning of Effective Dynamics (G-LED), instances of high dimensional data are down sampled to a lower dimensional manifold that is evolved through an auto-regressive attention mechanism. In turn, Bayesian diffusion models, that map this low-dimensional manifold onto its corresponding high-dimensional space, capture the statistics of the system dynamics. We demonstrate the capabilities and drawbacks of G-LED in simulations of several benchmark systems, including the Kuramoto-Sivashinsky (KS) equation, two-dimensional high Reynolds number flow over a backward-facing step, and simulations of three-dimensional turbulent channel flow. The results demonstrate that generative learning offers new frontiers for the accurate forecasting of the statistical properties of complex systems at a reduced computational cost.

Michal Balcerak, Tamaz Amiranashvili, Antonio Terpin et al.

Current state-of-the-art generative models map noise to data distributions by matching flows or scores. A key limitation of these models is their inability to readily integrate available partial observations and additional priors. In contrast, energy-based models (EBMs) address this by incorporating corresponding scalar energy terms. Here, we propose Energy Matching, a framework that endows flow-based approaches with the flexibility of EBMs. Far from the data manifold, samples move from noise to data along irrotational, optimal transport paths. As they approach the data manifold, an entropic energy term guides the system into a Boltzmann equilibrium distribution, explicitly capturing the underlying likelihood structure of the data. We parameterize these dynamics with a single time-independent scalar field, which serves as both a powerful generator and a flexible prior for effective regularization of inverse problems. The present method substantially outperforms existing EBMs on CIFAR-10 and ImageNet generation in terms of fidelity, while retaining simulation-free training of transport-based approaches away from the data manifold. Furthermore, we leverage the flexibility of the method to introduce an interaction energy that supports the exploration of diverse modes, which we demonstrate in a controlled protein generation setting. This approach learns a scalar potential energy, without time conditioning, auxiliary generators, or additional networks, marking a significant departure from recent EBM methods. We believe this simplified yet rigorous formulation significantly advances EBMs capabilities and paves the way for their wider adoption in generative modeling in diverse domains.

Lucas Amoudruz, Sergey Litvinov, Petros Koumoutsakos

Biomedical applications such as targeted drug delivery, microsurgery, and sensing rely on reaching precise areas within the body in a minimally invasive way. Artificial bacterial flagella (ABFs) have emerged as potential tools for this task by navigating through the circulatory system with the help of external magnetic fields. While their swimming characteristics are well understood in simple settings, their controlled navigation through realistic capillary networks remains a significant challenge due to the complexity of blood flow and the high computational cost of detailed simulations. We address this challenge by conducting numerical simulations of ABFs in retinal capillaries, propelled by an external magnetic field. The simulations are based on a validated blood model that predicts the dynamics of individual red blood cells and their hydrodynamic interactions with ABFs. The magnetic field follows a control policy that brings the ABF to a prescribed target. The control policy is learned with an actor-critic, off-policy reinforcement learning algorithm coupled with a reduced-order model of the system. We show that the same policy robustly guides the ABF to a prescribed target in both the reduced-order model and the fine-grained blood simulations. This approach is suitable for designing robust control policies for personalized medicine at moderate computational cost.

Jan-Philipp von Bassewitz, Sebastian Kaltenbach, Petros Koumoutsakos

Reliable predictions of critical phenomena, such as weather, wildfires and epidemics often rely on models described by Partial Differential Equations (PDEs). However, simulations that capture the full range of spatio-temporal scales described by such PDEs are often prohibitively expensive. Consequently, coarse-grained simulations are usually deployed that adopt various heuristics and empirical closure terms to account for the missing information. We propose a novel and systematic approach for identifying closures in under-resolved PDEs using grid-based Reinforcement Learning. This formulation incorporates inductive bias and exploits locality by deploying a central policy represented efficiently by a Fully Convolutional Network (FCN). We demonstrate the capabilities and limitations of our framework through numerical solutions of the advection equation and the Burgers' equation. Our results show accurate predictions for in- and out-of-distribution test cases as well as a significant speedup compared to resolving all scales.

Lucas Amoudruz, Petr Karnakov, Petros Koumoutsakos

Contactless manipulation of small objects is essential for biomedical and chemical applications, such as cell analysis, assisted fertilisation, and precision chemistry. Established methods, including optical, acoustic, and magnetic tweezers, are now complemented by flow control techniques that use flow-induced motion to enable precise and versatile manipulation. However, trapping multiple particles in fluid remains a challenge. This study introduces a novel control algorithm capable of steering multiple particles in flow. The system uses rotating disks to generate flow fields that transport particles to precise locations. Disk rotations are governed by a feedback control policy based on the Optimising a Discrete Loss (ODIL) framework, which combines fluid dynamics equations with path objectives into a single loss function. Our experiments, conducted in both simulations and with the physical device, demonstrate the capability of the approach to transport two beads simultaneously to predefined locations, advancing robust contactless particle manipulation for biomedical applications.

Lucas Amoudruz, Sergey Litvinov, Costas Papadimitriou et al.

Inverse problems are crucial for many applications in science, engineering and medicine that involve data assimilation, design, and imaging. Their solution infers the parameters or latent states of a complex system from noisy data and partially observable processes. When measurements are an incomplete or indirect view of the system, additional knowledge is required to accurately solve the inverse problem. Adopting a physical model of the system in the form of partial differential equations (PDEs) is a potent method to close this gap. In particular, the method of optimizing a discrete loss (ODIL) has shown great potential in terms of robustness and computational cost. In this work, we introduce B-ODIL, a Bayesian extension of ODIL, that integrates the PDE loss of ODIL as prior knowledge and combines it with a likelihood describing the data. B-ODIL employs a Bayesian formulation of PDE-based inverse problems to infer solutions with quantified uncertainties. We demonstrate the capabilities of B-ODIL in a series of synthetic benchmarks involving PDEs in one, two, and three dimensions. We showcase the application of B-ODIL in estimating tumor concentration and its uncertainty in a patient's brain from MRI scans using a three-dimensional tumor growth model.

Vasileios Saketos, Sebastian Kaltenbach, Sergey Litvinov et al.

Algorithmic discovery has traditionally relied on human ingenuity and extensive experimentation. Here we investigate whether a prominent scientific computing algorithm, the Kalman Filter, can be discovered through an automated, data-driven, evolutionary process that relies on Cartesian Genetic Programming (CGP) and Large Language Models (LLM). We evaluate the contributions of both modalities (CGP and LLM) in discovering the Kalman filter under varying conditions. Our results demonstrate that our framework of CGP and LLM-assisted evolution converges to near-optimal solutions when Kalman optimality assumptions hold. When these assumptions are violated, our framework evolves interpretable alternatives that outperform the Kalman filter. These results demonstrate that combining evolutionary algorithms and generative models for interpretable, data-driven synthesis of simple computational modules is a potent approach for algorithmic discovery in scientific computing.

Abhinav Singh, Petros Koumoutsakos

Topological defects in active polar fluids exhibit complex dynamics driven by internally generated stresses, reflecting the deep interplay between topology, flow, and non-equilibrium hydrodynamics. Feedback control offers a powerful means to guide such systems, enabling transitions between dynamic states. We investigated closed-loop steering of integer-charged defects in a confined active fluid by modulating the spatial profile of activity. Using a continuum hydrodynamic model, we show that localized control of active stress induces flow fields that can reposition and direct defects along prescribed trajectories by exploiting non-linear couplings in the system. A reinforcement learning framework is used to discover effective control strategies that produce robust defect transport across both trained and novel trajectories. The results highlight how AI agents can learn the underlying dynamics and spatially structure activity to manipulate topological excitations, offering insights into the controllability of active matter and the design of adaptive, self-organized materials.

Paul Fischer, Sebastian Kaltenbach, Sergey Litvinov et al.

The Lattice Boltzmann method (LBM) offers a powerful and versatile approach to simulating diverse hydrodynamic phenomena, spanning microfluidics to aerodynamics. The vast range of spatiotemporal scales inherent in these systems currently renders full resolution impractical, necessitating the development of effective closure models for under-resolved simulations. Under-resolved LBMs are unstable, and while there is a number of important efforts to stabilize them, they often face limitations in generalizing across scales and physical systems. We present a novel, data-driven, multiagent reinforcement learning (MARL) approach that drastically improves stability and accuracy of coarse-grained LBM simulations. The proposed method uses a convolutional neural network to dynamically control the local relaxation parameter for the LB across the simulation grid. The LB-MARL framework is showcased in turbulent Kolmogorov flows. We find that the MARL closures stabilize the simulations and recover the energy spectra of significantly more expensive fully resolved simulations while maintaining computational efficiency. The learned closure model can be transferred to flow scenarios unseen during training and has improved robustness and spectral accuracy compared to traditional LBM models. We believe that MARL closures open new frontiers for efficient and accurate simulations of a multitude of complex problems not accessible to present-day LB methods alone.

Han Gao, Sebastian Kaltenbach, Petros Koumoutsakos

Modeling and simulation of complex fluid flows with dynamics that span multiple spatio-temporal scales is a fundamental challenge in many scientific and engineering domains. Full-scale resolving simulations for systems such as highly turbulent flows are not feasible in the foreseeable future, and reduced-order models must capture dynamics that involve interactions across scales. In the present work, we propose a novel framework, Graph-based Learning of Effective Dynamics (Graph-LED), that leverages graph neural networks (GNNs), as well as an attention-based autoregressive model, to extract the effective dynamics from a small amount of simulation data. GNNs represent flow fields on unstructured meshes as graphs and effectively handle complex geometries and non-uniform grids. The proposed method combines a GNN based, dimensionality reduction for variable-size unstructured meshes with an autoregressive temporal attention model that can learn temporal dependencies automatically. We evaluated the proposed approach on a suite of fluid dynamics problems, including flow past a cylinder and flow over a backward-facing step over a range of Reynolds numbers. The results demonstrate robust and effective forecasting of spatio-temporal physics; in the case of the flow past a cylinder, both small-scale effects that occur close to the cylinder as well as its wake are accurately captured.

Daniel Waelchli, Pascal Weber, Petros Koumoutsakos

The discovery of individual objectives in collective behavior of complex dynamical systems such as fish schools and bacteria colonies is a long-standing challenge. Inverse reinforcement learning is a potent approach for addressing this challenge but its applicability to dynamical systems, involving continuous state-action spaces and multiple interacting agents, has been limited. In this study, we tackle this challenge by introducing an off-policy inverse multi-agent reinforcement learning algorithm (IMARL). Our approach combines the ReF-ER techniques with guided cost learning. By leveraging demonstrations, our algorithm automatically uncovers the reward function and learns an effective policy for the agents. Through extensive experimentation, we demonstrate that the proposed policy captures the behavior observed in the provided data, and achieves promising results across problem domains including single agent models in the OpenAI gym and multi-agent models of schooling behavior. The present study shows that the proposed IMARL algorithm is a significant step towards understanding collective dynamics from the perspective of its constituents, and showcases its value as a tool for studying complex physical systems exhibiting collective behaviour.

H. Jane Bae, Petros Koumoutsakos

The predictive capabilities of turbulent flow simulations, critical for aerodynamic design and weather prediction, hinge on the choice of turbulence models. The abundance of data from experiments and simulations and the advent of machine learning have provided a boost to turbulence modeling efforts. However, simulations of turbulent flows remain hindered by the inability of heuristics and supervised learning to model the near-wall dynamics. We address this challenge by introducing scientific multi-agent reinforcement learning (SciMARL) for the discovery of wall models for large-eddy simulations (LES). In SciMARL, discretization points act also as cooperating agents that learn to supply the LES closure model. The agents self-learn using limited data and generalize to extreme Reynolds numbers and previously unseen geometries. The present simulations reduce by several orders of magnitude the computational cost over fully-resolved simulations while reproducing key flow quantities. We believe that SciMARL creates unprecedented capabilities for the simulation of turbulent flows.

Ioannis Mandralis, Pascal Weber, Guido Novati et al.

Swimming organisms can escape their predators by creating and harnessing unsteady flow fields through their body motions. Stochastic optimization and flow simulations have identified escape patterns that are consistent with those observed in natural larval swimmers. However, these patterns have been limited by the specification of a particular cost function and depend on a prescribed functional form of the body motion. Here, we deploy reinforcement learning to discover swimmer escape patterns for larval fish under energy constraints. The identified patterns include the C-start mechanism, in addition to more energetically efficient escapes. We find that maximizing distance with limited energy requires swimming via short bursts of accelerating motion interlinked with phases of gliding. The present, data efficient, reinforcement learning algorithm results in an array of patterns that reveal practical flow optimization principles for efficient swimming and the methodology can be transferred to the control of aquatic robotic devices operating under energy constraints.

Peter Gunnarson, Ioannis Mandralis, Guido Novati et al.

Efficient point-to-point navigation in the presence of a background flow field is important for robotic applications such as ocean surveying. In such applications, robots may only have knowledge of their immediate surroundings or be faced with time-varying currents, which limits the use of optimal control techniques for planning trajectories. Here, we apply a novel Reinforcement Learning algorithm to discover time-efficient navigation policies to steer a fixed-speed swimmer through an unsteady two-dimensional flow field. The algorithm entails inputting environmental cues into a deep neural network that determines the swimmer's actions, and deploying Remember and Forget Experience replay. We find that the resulting swimmers successfully exploit the background flow to reach the target, but that this success depends on the type of sensed environmental cue. Surprisingly, a velocity sensing approach outperformed a bio-mimetic vorticity sensing approach by nearly two-fold in success rate. Equipped with local velocity measurements, the reinforcement learning algorithm achieved near 100% success in reaching the target locations while approaching the time-efficiency of paths found by a global optimal control planner.

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.

Guido Novati, Hugues Lascombes de Laroussilhe, Petros Koumoutsakos

The modeling of turbulent flows is critical to scientific and engineering problems ranging from aircraft design to weather forecasting and climate prediction. Over the last sixty years numerous turbulence models have been proposed, largely based on physical insight and engineering intuition. Recent advances in machine learning and data science have incited new efforts to complement these approaches. To date, all such efforts have focused on supervised learning which, despite demonstrated promise, encounters difficulties in generalizing beyond the distributions of the training data. In this work we introduce multi-agent reinforcement learning (MARL) as an automated discovery tool of turbulence models. We demonstrate the potential of this approach on Large Eddy Simulations of homogeneous and isotropic turbulence using as reward the recovery of the statistical properties of Direct Numerical Simulations. Here, the closure model is formulated as a control policy enacted by cooperating agents, which detect critical spatio-temporal patterns in the flow field to estimate the unresolved sub-grid scale (SGS) physics. The present results are obtained with state-of-the-art algorithms based on experience replay and compare favorably with established dynamic SGS modeling approaches. Moreover, we show that the present turbulence models generalize across grid sizes and flow conditions as expressed by the Reynolds numbers.

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.

Steven Brunton, Bernd Noack, Petros Koumoutsakos

The field of fluid mechanics is rapidly advancing, driven by unprecedented volumes of data from field measurements, experiments and large-scale simulations at multiple spatiotemporal scales. Machine learning offers a wealth of techniques to extract information from data that could be translated into knowledge about the underlying fluid mechanics. Moreover, machine learning algorithms can augment domain knowledge and automate tasks related to flow control and optimization. This article presents an overview of past history, current developments, and emerging opportunities of machine learning for fluid mechanics. It outlines fundamental machine learning methodologies and discusses their uses for understanding, modeling, optimizing, and controlling fluid flows. The strengths and limitations of these methods are addressed from the perspective of scientific inquiry that considers data as an inherent part of modeling, experimentation, and simulation. Machine learning provides a powerful information processing framework that can enrich, and possibly even transform, current lines of fluid mechanics research and industrial applications.

Guido Novati, Petros Koumoutsakos

Experience replay (ER) is a fundamental component of off-policy deep reinforcement learning (RL). ER recalls experiences from past iterations to compute gradient estimates for the current policy, increasing data-efficiency. However, the accuracy of such updates may deteriorate when the policy diverges from past behaviors and can undermine the performance of ER. Many algorithms mitigate this issue by tuning hyper-parameters to slow down policy changes. An alternative is to actively enforce the similarity between policy and the experiences in the replay memory. We introduce Remember and Forget Experience Replay (ReF-ER), a novel method that can enhance RL algorithms with parameterized policies. ReF-ER (1) skips gradients computed from experiences that are too unlikely with the current policy and (2) regulates policy changes within a trust region of the replayed behaviors. We couple ReF-ER with Q-learning, deterministic policy gradient and off-policy gradient methods. We find that ReF-ER consistently improves the performance of continuous-action, off-policy RL on fully observable benchmarks and partially observable flow control problems.

Guido Novati, Lakshminarayanan Mahadevan, Petros Koumoutsakos

Controlled gliding is one of the most energetically efficient modes of transportation for natural and human powered fliers. Here we demonstrate that gliding and landing strategies with different optimality criteria can be identified through deep reinforcement learning without explicit knowledge of the underlying physics. We combine a two dimensional model of a controlled elliptical body with deep reinforcement learning (D-RL) to achieve gliding with either minimum energy expenditure, or fastest time of arrival, at a predetermined location. In both cases the gliding trajectories are smooth, although energy/time optimal strategies are distinguished by small/high frequency actuations. We examine the effects of the ellipse's shape and weight on the optimal policies for controlled gliding. Surprisingly, we find that the model-free reinforcement learning leads to more robust gliding than model-based optimal control strategies with a modest additional computational cost. We also demonstrate that the gliders with D-RL can generalize their strategies to reach the target location from previously unseen starting positions. The model-free character and robustness of D-RL suggests a promising framework for developing mechanical devices capable of exploiting complex flow environments.

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.

Siddhartha Verma, Guido Novati, Petros Koumoutsakos

Fish in schooling formations navigate complex flow-fields replete with mechanical energy in the vortex wakes of their companions. Their schooling behaviour has been associated with evolutionary advantages including collective energy savings. How fish harvest energy from their complex fluid environment and the underlying physical mechanisms governing energy-extraction during collective swimming, is still unknown. Here we show that fish can improve their sustained propulsive efficiency by actively following, and judiciously intercepting, vortices in the wake of other swimmers. This swimming strategy leads to collective energy-savings and is revealed through the first ever combination of deep reinforcement learning with high-fidelity flow simulations. We find that a `smart-swimmer' can adapt its position and body deformation to synchronise with the momentum of the oncoming vortices, improving its average swimming-efficiency at no cost to the leader. The results show that fish may harvest energy deposited in vortices produced by their peers, and support the conjecture that swimming in formation is energetically advantageous. Moreover, this study demonstrates that deep reinforcement learning can produce navigation algorithms for complex flow-fields, with promising implications for energy savings in autonomous robotic swarms.

Wonmin Byeon, Qin Wang, Rupesh Kumar Srivastava et al.

Video prediction models based on convolutional networks, recurrent networks, and their combinations often result in blurry predictions. We identify an important contributing factor for imprecise predictions that has not been studied adequately in the literature: blind spots, i.e., lack of access to all relevant past information for accurately predicting the future. To address this issue, we introduce a fully context-aware architecture that captures the entire available past context for each pixel using Parallel Multi-Dimensional LSTM units and aggregates it using blending units. Our model outperforms a strong baseline network of 20 recurrent convolutional layers and yields state-of-the-art performance for next step prediction on three challenging real-world video datasets: Human 3.6M, Caltech Pedestrian, and UCF-101. Moreover, it does so with fewer parameters than several recently proposed models, and does not rely on deep convolutional networks, multi-scale architectures, separation of background and foreground modeling, motion flow learning, or adversarial training. These results highlight that full awareness of past context is of crucial importance for video prediction.