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Modelling of Underwater Vehicles using Physics-Informed Neural Networks with Control

Abdelhakim Amer, David Felsager, Yury Brodskiy, Andriy Sarabakha

TL;DR

This paper addresses the challenge of accurately modelling underwater vehicle dynamics for control by introducing PINC, a physics-informed neural network with control that extends PINNs to incorporate control inputs and autoregressive rollouts. The approach uses a residual integration framework where the state at time $T$ satisfies $\mathbf{x}(T) \approx \mathbf{x}(0) + \mathcal{N}(\mathbf{x}(0),\mathbf{u}(0),T)$, trained with a combination of data losses and physics-based regularization across multiple loss terms. A detailed neural architecture with yaw re-parameterization, rotational structure, and layer normalization is paired with a suite of loss functions ($\mathcal{L}_D$, $\mathcal{L}_P$, $\mathcal{L}_{IC}$, $\mathcal{L}_R$, $\mathcal{L}_{PR}$) and gradient-weighting schemes to optimize long-horizon predictive accuracy. Validation on a simulated underwater vehicle shows that PINC outperforms a non-physics baseline in long-horizon predictions, while being computationally efficient and robust to input noise. The results suggest strong potential for real-world deployment in MPC and online adaptation, with future work extending to full 3D rotational dynamics and real trajectory data.

Abstract

Physics-informed neural networks (PINNs) integrate physical laws with data-driven models to improve generalization and sample efficiency. This work introduces an open-source implementation of the Physics-Informed Neural Network with Control (PINC) framework, designed to model the dynamics of an underwater vehicle. Using initial states, control actions, and time inputs, PINC extends PINNs to enable physically consistent transitions beyond the training domain. Various PINC configurations are tested, including differing loss functions, gradient-weighting schemes, and hyperparameters. Validation on a simulated underwater vehicle demonstrates more accurate long-horizon predictions compared to a non-physics-informed baseline

Modelling of Underwater Vehicles using Physics-Informed Neural Networks with Control

TL;DR

This paper addresses the challenge of accurately modelling underwater vehicle dynamics for control by introducing PINC, a physics-informed neural network with control that extends PINNs to incorporate control inputs and autoregressive rollouts. The approach uses a residual integration framework where the state at time satisfies , trained with a combination of data losses and physics-based regularization across multiple loss terms. A detailed neural architecture with yaw re-parameterization, rotational structure, and layer normalization is paired with a suite of loss functions (, , , , ) and gradient-weighting schemes to optimize long-horizon predictive accuracy. Validation on a simulated underwater vehicle shows that PINC outperforms a non-physics baseline in long-horizon predictions, while being computationally efficient and robust to input noise. The results suggest strong potential for real-world deployment in MPC and online adaptation, with future work extending to full 3D rotational dynamics and real trajectory data.

Abstract

Physics-informed neural networks (PINNs) integrate physical laws with data-driven models to improve generalization and sample efficiency. This work introduces an open-source implementation of the Physics-Informed Neural Network with Control (PINC) framework, designed to model the dynamics of an underwater vehicle. Using initial states, control actions, and time inputs, PINC extends PINNs to enable physically consistent transitions beyond the training domain. Various PINC configurations are tested, including differing loss functions, gradient-weighting schemes, and hyperparameters. Validation on a simulated underwater vehicle demonstrates more accurate long-horizon predictions compared to a non-physics-informed baseline
Paper Structure (33 sections, 19 equations, 7 figures)

This paper contains 33 sections, 19 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Related Works
  3. Problem Formulation
  4. Proposed Method
  5. PINC: Physics-Informed Neural Network with Control
  6. Neural Network Architecture
  7. Residual Formulation for ODE Integration
  8. Layers, Neurons, and Activations
  9. State Re-Parameterization for Yaw
  10. Rotational Structural Information
  11. Layer Normalization
  12. Loss Functions
  13. One-step-ahead Prediction Loss
  14. Physics Loss
  15. Initial Condition Loss
  16. ...and 18 more sections

Figures (7)

  • Figure 1: Coordinate frame of the ROV with respect to the world frame.
  • Figure 2: PINC model output used to calculate the MSE data loss by comparing it to the ground truth state and used with AD and the underlying physics function to find the MSE physics loss.
  • Figure 3: Neural network size.
  • Figure 7: Effect of adding rollout loss vs. physics loss on VPT: Incorporating rollout loss with physics loss does not yield any additional performance gains compared to using physics loss alone, as shown in the Loss and VPT metrics.
  • Figure 8: Effect of noise on learning: Noise was introduced into the ROV simulation model to evaluate the learning performance. Incorporating physics information significantly reduces underfitting and enhances robustness to noise, enabling more reliable learning.
  • ...and 2 more figures

Theorems & Definitions (5)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5