Using the SEKF to Transfer NN Models of Dynamical Systems with Limited Data

Deep state-space models (DSSMs) enable temporal predictions by learning the underlying dynamics of observed sequence data. They are often trained by maximising the evidence lower bound. However, as we show, this does not ensure the model actually learns the underlying dynamics. We therefore propose a constrained optimisation framework as a general approach for training DSSMs. Building upon this, we introduce the extended Kalman VAE (EKVAE), which combines amortised variational inference with classic Bayesian filtering/smoothing to model dynamics more accurately than RNN-based DSSMs. Our results show that the constrained optimisation framework significantly improves system identification and prediction accuracy on the example of established state-of-the-art DSSMs. The EKVAE outperforms previous models w.r.t. prediction accuracy, achieves remarkable results in identifying dynamical systems, and can furthermore successfully learn state-space representations where static and dynamic features are disentangled.

Model-based design of experiments (MBDOE) is essential for efficient parameter estimation in nonlinear dynamical systems. However, conventional adaptive MBDOE requires costly posterior inference and design optimization between each experimental step, precluding real-time applications. We address this by combining Deep Adaptive Design (DAD), which amortizes sequential design into a neural network policy trained offline, with differentiable mechanistic models. For dynamical systems with known governing equations but uncertain parameters, we extend sequential contrastive training objectives to handle nuisance parameters and propose a transformer-based policy architecture that respects the temporal structure of dynamical systems. We demonstrate the approach on four systems of increasing complexity: a fed-batch bioreactor with Monod kinetics, a Haldane bioreactor with uncertain substrate inhibition, a two-compartment pharmacokinetic model with nuisance clearance parameters, and a DC motor for real-time deployment.

Learning system dynamics from observations is a critical problem in many applications over various real-world complex systems, e.g., climate, ecology, and fluid systems. Recently, neural dynamics modeling method have become a prevalent solution that embeds the object's observations into a latent space before learning dynamics using neural methods such as neural Ordinary Differential Equations (ODE). Existing dynamics modeling methods induce a specific model for each observation of different complex systems, resulting in poor generalization across systems. Inspired by the great success of pre-trained models, we conduct a generalized Pre-trained Dynamics EncoDER (PDEDER) which can embed the original state observations into a latent space where the dynamics can be captured more easily. To conduct the generalized PDEDER, we pre-train any Pre-trained Language Model (PLM) by minimizing the Lyapunov exponent objective, which constrains the chaotic behavior of governing dynamics learned in the latent space. By penalizing the divergence of embedded observations, our PDEDER promotes locally stable and well-structured latent dynamics, thereby facilitating more effective dynamics modeling than in the original observation space. In addition, we incorporate reconstruction and forecasting objectives to mitigate the risk of obtaining an over-smoothed latent space. Specifically, we collect 152 sets of real-world and synthetic observations from 23 complex systems as pre-training corpora and employ them to pre-train PDEDER. Given any future dynamic observation, we can fine-tune PDEDER with any specific dynamics modeling method. We evaluate PDEDER on 12 dynamic systems by short/long-term forecasting under both in-domain and cross-domain settings, and the empirical results indicate the effectiveness and generalizability of PDEDER.