Data-driven recurrent models to forecast and control high-dimensional dynamical systems

Abstract

Many dynamical systems in science and engineering are high-dimensional, nonlinear, and only partially observable. Moreover, governing equations may be unknown and only limited observational data may be available. This dissertation develops recurrent neural network (RNN) methods that address three ubiquitous challenges in this setting: (i) state estimation from sparse sensors, (ii) data-driven surrogate modeling for control, and (iii) autonomous forecasting of high-dimensional dynamics. The key enabler throughout this work is the construction of recurrent latent structure. The recurrent architecture of RNNs provides an inductive bias well-matched to temporal dynamics, encoding high-dimensional nonlinear physics more expressively than linear projection-based techniques while learning far more data-efficiently than modern attention-based architectures such as transformers and foundation models. This work develops temporally embedded shallow decoder and echo state network (ESN) architectures that enable high-dimensional estimation, control, and forecasting from partial, sparse sensor observations, with applications to turbulent fluid flows and chaotic dynamical systems. The SHallow REcurrent Decoder (SHRED) reconstructs high-dimensional states from time histories of sparse sensor measurements. By utilizing sequential embeddings of sensor measurements, SHRED leverages Takens' theorem and incorporates the temporal context that is absent from methods that reconstruct from static measurements alone, such as gappy POD. Its nonlinear decoder also enables enhanced representation and compression capability. ESNs, a form of reservoir computer, are designed as surrogate models within model predictive control loops, demonstrated on lift control of flow past a cylinder and forecasting of chaotic dynamical systems. ESNs are orders of magnitude cheaper to train than standard RNN models, while remaining sufficiently lightweight to be embedded within optimization loops that require thousands of model evaluations per control step. To support reproducibility and community adoption, we develop OpenReservoirComputing, an open source, GPU accelerated reservoir computing library implemented in JAX. As one of the only autodiff-native software packages for ESNs, OpenReservoirComputing directly enables the use of ESNs within larger neural network models, as well as optimization and control loops. Finally, the two methods are coupled to construct a reduced order model capable of forecasting directly from sparse sensor streams, following an offline training stage with full state information. SHRED maps sensor measurements to a learned latent representation, and the ESN propagates this latent state forward in time while allowing for the online assimilation of sensor measurements. This closes the loop between sensing and modeling, with preliminary results suggesting that the learned latent representation captures key geometric and statistical structure of the underlying attractor.