SALSA-RL: Stability Analysis in the Latent Space of Actions for Reinforcement Learning

Modern deep reinforcement learning (DRL) methods have made significant advances in handling continuous action spaces. However, real-world control systems--especially those requiring precise and reliable performance--often demand formal stability, and existing DRL approaches typically lack explicit mechanisms to ensure or analyze stability. To address this limitation, we propose SALSA-RL (Stability Analysis in the Latent Space of Actions), a novel RL framework that models control actions as dynamic, time-dependent variables evolving within a latent space. By employing a pre-trained encoder-decoder and a state-dependent linear system, our approach enables both stability analysis and interpretability. We demonstrated that SALSA-RL can be deployed in a non-invasive manner for assessing the local stability of actions from pretrained RL agents without compromising on performance across diverse benchmark environments. By enabling a more interpretable analysis of action generation, SALSA-RL provides a powerful tool for advancing the design, analysis, and theoretical understanding of RL systems.

CondensNet: Enabling stable long-term climate simulations via hybrid deep learning models with adaptive physical constraints

Accurate and efficient climate simulations are crucial for understanding Earth's evolving climate. However, current general circulation models (GCMs) face challenges in capturing unresolved physical processes, such as cloud and convection. A common solution is to adopt cloud resolving models, that provide more accurate results than the standard subgrid parametrisation schemes typically used in GCMs. However, cloud resolving models, also referred to as super paramtetrizations, remain computationally prohibitive. Hybrid modeling, which integrates deep learning with equation-based GCMs, offers a promising alternative but often struggles with long-term stability and accuracy issues. In this work, we find that water vapor oversaturation during condensation is a key factor compromising the stability of hybrid models. To address this, we introduce CondensNet, a novel neural network architecture that embeds a self-adaptive physical constraint to correct unphysical condensation processes. CondensNet effectively mitigates water vapor oversaturation, enhancing simulation stability while maintaining accuracy and improving computational efficiency compared to super parameterization schemes. We integrate CondensNet into a GCM to form PCNN-GCM (Physics-Constrained Neural Network GCM), a hybrid deep learning framework designed for long-term stable climate simulations in real-world conditions, including ocean and land. PCNN-GCM represents a significant milestone in hybrid climate modeling, as it shows a novel way to incorporate physical constraints adaptively, paving the way for accurate, lightweight, and stable long-term climate simulations.

Semi-Implicit Neural Ordinary Differential Equations

Classical neural ODEs trained with explicit methods are intrinsically limited by stability, crippling their efficiency and robustness for stiff learning problems that are common in graph learning and scientific machine learning. We present a semi-implicit neural ODE approach that exploits the partitionable structure of the underlying dynamics. Our technique leads to an implicit neural network with significant computational advantages over existing approaches because of enhanced stability and efficient linear solves during time integration. We show that our approach outperforms existing approaches on a variety of applications including graph classification and learning complex dynamical systems. We also demonstrate that our approach can train challenging neural ODEs where both explicit methods and fully implicit methods are intractable.

Improved deep learning of chaotic dynamical systems with multistep penalty losses

Predicting the long-term behavior of chaotic systems remains a formidable challenge due to their extreme sensitivity to initial conditions and the inherent limitations of traditional data-driven modeling approaches. This paper introduces a novel framework that addresses these challenges by leveraging the recently proposed multi-step penalty (MP) optimization technique. Our approach extends the applicability of MP optimization to a wide range of deep learning architectures, including Fourier Neural Operators and UNETs. By introducing penalized local discontinuities in the forecast trajectory, we effectively handle the non-convexity of loss landscapes commonly encountered in training neural networks for chaotic systems. We demonstrate the effectiveness of our method through its application to two challenging use-cases: the prediction of flow velocity evolution in two-dimensional turbulence and ocean dynamics using reanalysis data. Our results highlight the potential of this approach for accurate and stable long-term prediction of chaotic dynamics, paving the way for new advancements in data-driven modeling of complex natural phenomena.

Scalable and Consistent Graph Neural Networks for Distributed Mesh-based Data-driven Modeling

This work develops a distributed graph neural network (GNN) methodology for mesh-based modeling applications using a consistent neural message passing layer. As the name implies, the focus is on enabling scalable operations that satisfy physical consistency via halo nodes at sub-graph boundaries. Here, consistency refers to the fact that a GNN trained and evaluated on one rank (one large graph) is arithmetically equivalent to evaluations on multiple ranks (a partitioned graph). This concept is demonstrated by interfacing GNNs with NekRS, a GPU-capable exascale CFD solver developed at Argonne National Laboratory. It is shown how the NekRS mesh partitioning can be linked to the distributed GNN training and inference routines, resulting in a scalable mesh-based data-driven modeling workflow. We study the impact of consistency on the scalability of mesh-based GNNs, demonstrating efficient scaling in consistent GNNs for up to O(1B) graph nodes on the Frontier exascale supercomputer.

A competitive baseline for deep learning enhanced data assimilation using conditional Gaussian ensemble Kalman filtering

Ensemble Kalman Filtering (EnKF) is a popular technique for data assimilation, with far ranging applications. However, the vanilla EnKF framework is not well-defined when perturbations are nonlinear. We study two non-linear extensions of the vanilla EnKF - dubbed the conditional-Gaussian EnKF (CG-EnKF) and the normal score EnKF (NS-EnKF) - which sidestep assumptions of linearity by constructing the Kalman gain matrix with the `conditional Gaussian' update formula in place of the traditional one. We then compare these models against a state-of-the-art deep learning based particle filter called the score filter (SF). This model uses an expensive score diffusion model for estimating densities and also requires a strong assumption on the perturbation operator for validity. In our comparison, we find that CG-EnKF and NS-EnKF dramatically outperform SF for a canonical problem in high-dimensional multiscale data assimilation given by the Lorenz-96 system. Our analysis also demonstrates that the CG-EnKF and NS-EnKF can handle highly non-Gaussian additive noise perturbations, with the latter typically outperforming the former.

Measure-Theoretic Time-Delay Embedding

The celebrated Takens' embedding theorem provides a theoretical foundation for reconstructing the full state of a dynamical system from partial observations. However, the classical theorem assumes that the underlying system is deterministic and that observations are noise-free, limiting its applicability in real-world scenarios. Motivated by these limitations, we rigorously establish a measure-theoretic generalization that adopts an Eulerian description of the dynamics and recasts the embedding as a pushforward map between probability spaces. Our mathematical results leverage recent advances in optimal transportation theory. Building on our novel measure-theoretic time-delay embedding theory, we have developed a new computational framework that forecasts the full state of a dynamical system from time-lagged partial observations, engineered with better robustness to handle sparse and noisy data. We showcase the efficacy and versatility of our approach through several numerical examples, ranging from the classic Lorenz-63 system to large-scale, real-world applications such as NOAA sea surface temperature forecasting and ERA5 wind field reconstruction.

Mesh-based Super-Resolution of Fluid Flows with Multiscale Graph Neural Networks

A graph neural network (GNN) approach is introduced in this work which enables mesh-based three-dimensional super-resolution of fluid flows. In this framework, the GNN is designed to operate not on the full mesh-based field at once, but on localized meshes of elements (or cells) directly. To facilitate mesh-based GNN representations in a manner similar to spectral (or finite) element discretizations, a baseline GNN layer (termed a message passing layer, which updates local node properties) is modified to account for synchronization of coincident graph nodes, rendering compatibility with commonly used element-based mesh connectivities. The architecture is multiscale in nature, and is comprised of a combination of coarse-scale and fine-scale message passing layer sequences (termed processors) separated by a graph unpooling layer. The coarse-scale processor embeds a query element (alongside a set number of neighboring coarse elements) into a single latent graph representation using coarse-scale synchronized message passing over the element neighborhood, and the fine-scale processor leverages additional message passing operations on this latent graph to correct for interpolation errors. Demonstration studies are performed using hexahedral mesh-based data from Taylor-Green Vortex flow simulations at Reynolds numbers of 1600 and 3200. Through analysis of both global and local errors, the results ultimately show how the GNN is able to produce accurate super-resolved fields compared to targets in both coarse-scale and multiscale model configurations. Reconstruction errors for fixed architectures were found to increase in proportion to the Reynolds number, while the inclusion of surrounding coarse element neighbors was found to improve predictions at Re=1600, but not at Re=3200.

Higher order quantum reservoir computing for non-intrusive reduced-order models

Forecasting dynamical systems is of importance to numerous real-world applications. When possible, dynamical systems forecasts are constructed based on first-principles-based models such as through the use of differential equations. When these equations are unknown, non-intrusive techniques must be utilized to build predictive models from data alone. Machine learning (ML) methods have recently been used for such tasks. Moreover, ML methods provide the added advantage of significant reductions in time-to-solution for predictions in contrast with first-principle based models. However, many state-of-the-art ML-based methods for forecasting rely on neural networks, which may be expensive to train and necessitate requirements for large amounts of memory. In this work, we propose a quantum mechanics inspired ML modeling strategy for learning nonlinear dynamical systems that provides data-driven forecasts for complex dynamical systems with reduced training time and memory costs. This approach, denoted the quantum reservoir computing technique (QRC), is a hybrid quantum-classical framework employing an ensemble of interconnected small quantum systems via classical linear feedback connections. By mapping the dynamical state to a suitable quantum representation amenable to unitary operations, QRC is able to predict complex nonlinear dynamical systems in a stable and accurate manner. We demonstrate the efficacy of this framework through benchmark forecasts of the NOAA Optimal Interpolation Sea Surface Temperature dataset and compare the performance of QRC to other ML methods.

Divide And Conquer: Learning Chaotic Dynamical Systems With Multistep Penalty Neural Ordinary Differential Equations

Forecasting high-dimensional dynamical systems is a fundamental challenge in various fields, such as the geosciences and engineering. Neural Ordinary Differential Equations (NODEs), which combine the power of neural networks and numerical solvers, have emerged as a promising algorithm for forecasting complex nonlinear dynamical systems. However, classical techniques used for NODE training are ineffective for learning chaotic dynamical systems. In this work, we propose a novel NODE-training approach that allows for robust learning of chaotic dynamical systems. Our method addresses the challenges of non-convexity and exploding gradients associated with underlying chaotic dynamics. Training data trajectories from such systems are split into multiple, non-overlapping time windows. In addition to the deviation from the training data, the optimization loss term further penalizes the discontinuities of the predicted trajectory between the time windows. The window size is selected based on the fastest Lyapunov time scale of the system. Multi-step penalty(MP) method is first demonstrated on Lorenz equation, to illustrate how it improves the loss landscape and thereby accelerating the optimization convergence. MP method can optimize chaotic systems in a manner similar to least-squares shadowing with significantly lower computational costs. Our proposed algorithm, denoted the Multistep Penalty NODE(MP-NODE), is applied to chaotic systems such as the Kuramoto-Sivashinsky equation and the two-dimensional Kolmogorov flow. It is observed that MP-NODE provide viable performance for such chaotic systems, not only for short-term trajectory predictions but also for invariant statistics that are hallmarks of the chaotic nature of these dynamics.