Data-driven discrete-time deep recurrent neural network-based modeling for dissipative systems

Accurate dynamical modeling is essential for simulation and control of embodied systems, yet first-principles models of electromechanical systems often fail to capture complex dissipative effects such as joint friction, stray losses, and structural damping. While residual-learning physics-informed neural networks (PINNs) can effectively augment imperfect first-principles models with data-driven components, the residual terms are typically implemented as unconstrained multilayer perceptrons (MLPs), which may inadvertently inject artificial energy into the system. To more faithfully model the dissipative dynamics, we propose DiLaR-PINN, a dissipative latent residual PINN designed to learn unmodeled dissipative effects in a physically consistent manner. Structurally, the residual network operates only on unmeasurable (latent) state components and is parameterized in a skew-dissipative form that guarantees non-increasing energy for any choice of network parameters. To enable stable and data-efficient training under partial measurability of the state, we further develop a recurrent rollout scheme with a curriculum-based sequence length extension strategy. We validate DiLaR-PINN on a real-world helicopter system and compare it against four baselines: a pure physical model (without a residual network), an unstructured residual MLP, a DiLaR variant with a soft dissipativity constraint, and a black-box LSTM. The results demonstrate that DiLaR-PINN more accurately captures dissipative effects and achieves superior long-horizon extrapolation performance.

State-space models are traditionally based on physical knowledge, but multi-step predictions from these physical models can be poor due to model inaccuracy. Black-box deep learning has shown promise as an alternative. However, these methods rely on the availability of large datasets and potentially available physical knowledge is neglected. We propose the PG-RSSNN, a physics-guided recurrent state-space neural network that incorporates recurrent structures to enable the use of non-saturating activation functions in multi-step prediction. It mitigates the vanishing gradients and eliminates the risk of numerical divergence in training seen in existing structures that feed back state estimates. Results across multiple systems with various physical model imperfections, from linear state-space models with Gaussian noise to a robotic arm and a cascaded water tank system, show that the proposed PG-RSSNN maintains stable training behavior, and improves multi-step predictions, as compared with black-box neural networks and physics-only models, even with limited training data and when physical models are only partially known.

We introduce \emph{Dynamical Physics-Modeled Neural Networks} (DynPMNNs), a continuous-time deep learning architecture in which each hidden layer is defined as the solution of an ordinary differential equation. Unlike classical feed-forward networks, this approach replaces static activation functions with time-evolving dynamical systems, providing a biologically inspired interpretation of hidden-layer behavior and enabling the integration of physically meaningful models. The framework is rigorously grounded in Reproducing Kernel Banach Spaces (RKBSs), allowing DynPMNNs to be characterized as finite-dimensional solutions of an abstract training problem and revealing structural connections with standard neural networks. We present a concrete implementation based on the FitzHugh--Nagumo model for neuronal activation, where numerical ODE solvers are embedded into the computational graph via Euler-type schemes. Both network weights and dynamical parameters are trained jointly. Through experiments on the California Housing dataset, we compare DynPMNNs with Neural ODEs (NODEs) and Closed-form Continuous-Time Networks (CfCs). Despite using fewer trainable parameters, DynPMNNs achieve competitive performance. These results position DynPMNNs as a principled bridge between dynamical systems and deep learning, with promising directions for further research in expressivity, stability, and physics-based modeling.