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Learning Physics Informed Neural ODEs With Partial Measurements

Paul Ghanem, Ahmet Demirkaya, Tales Imbiriba, Alireza Ramezani, Zachary Danziger, Deniz Erdogmus

TL;DR

This work tackles learning dynamics when part of the state is unmeasured by casting the problem as learning latent dynamics within a Physics-Informed Neural ODE framework. It introduces a sequential, alternating Newton optimization that iteratively estimates latent states and model parameters, aided by a physics-informed loss to constrain the unobserved dynamics. The approach yields state trajectories and parameters that outperform several baselines (NODE, NODE-LSTM, SPINODE, RM, PR-SSM) across both synthetic and real datasets, including Hodgkin-Huxley, cart-pole, and an electro-mechanical positioning system. By exploiting identifiability assumptions and leveraging a Kalman-filter–like recursion, the method enables accurate learning of hidden dynamics from partial measurements, with potential applicability to online data assimilation and scalable Newton-based training.

Abstract

Learning dynamics governing physical and spatiotemporal processes is a challenging problem, especially in scenarios where states are partially measured. In this work, we tackle the problem of learning dynamics governing these systems when parts of the system's states are not measured, specifically when the dynamics generating the non-measured states are unknown. Inspired by state estimation theory and Physics Informed Neural ODEs, we present a sequential optimization framework in which dynamics governing unmeasured processes can be learned. We demonstrate the performance of the proposed approach leveraging numerical simulations and a real dataset extracted from an electro-mechanical positioning system. We show how the underlying equations fit into our formalism and demonstrate the improved performance of the proposed method when compared with baselines.

Learning Physics Informed Neural ODEs With Partial Measurements

TL;DR

This work tackles learning dynamics when part of the state is unmeasured by casting the problem as learning latent dynamics within a Physics-Informed Neural ODE framework. It introduces a sequential, alternating Newton optimization that iteratively estimates latent states and model parameters, aided by a physics-informed loss to constrain the unobserved dynamics. The approach yields state trajectories and parameters that outperform several baselines (NODE, NODE-LSTM, SPINODE, RM, PR-SSM) across both synthetic and real datasets, including Hodgkin-Huxley, cart-pole, and an electro-mechanical positioning system. By exploiting identifiability assumptions and leveraging a Kalman-filter–like recursion, the method enables accurate learning of hidden dynamics from partial measurements, with potential applicability to online data assimilation and scalable Newton-based training.

Abstract

Learning dynamics governing physical and spatiotemporal processes is a challenging problem, especially in scenarios where states are partially measured. In this work, we tackle the problem of learning dynamics governing these systems when parts of the system's states are not measured, specifically when the dynamics generating the non-measured states are unknown. Inspired by state estimation theory and Physics Informed Neural ODEs, we present a sequential optimization framework in which dynamics governing unmeasured processes can be learned. We demonstrate the performance of the proposed approach leveraging numerical simulations and a real dataset extracted from an electro-mechanical positioning system. We show how the underlying equations fit into our formalism and demonstrate the improved performance of the proposed method when compared with baselines.

Paper Structure

This paper contains 26 sections, 1 theorem, 67 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Model and Background
  4. Model
  5. Identifiability of latent states:
  6. Discrete Generative model
  7. Method
  8. Sequential Newton Derivation
  9. Experiments
  10. Hodgkin-Huxley Neuron Model
  11. Cart-pole System
  12. Electro-mechanical positioning system
  13. Conclusions
  14. Proof of Theorem 1
  15. Vanishing gradients
  16. ...and 11 more sections

Key Result

Theorem 1

Given $\hat{\theta}(t_{i-1}) \in \hat{\Theta}_{i-1}$ and $\hat{x}(t_{i-1}) \in \hat{X}_{i-1}$, and known $P_{\theta_{i-1}}\in R^{d_{\theta}\times d_{\theta}}$ and $P_{x_{i-1}} \in R^{d_x \times d_x}$, the recursive equations for computing $\hat{x}(t_i)$ and $\hat{\theta}(t_i)$ that minimize (eq:div with $P_{\theta_i}^{-}$, $P_{x_i}^{-}$ being intermediate matrices and $P_{\theta_i}^{}$ and $P_{x

Figures (10)

  • Figure 1: The generative model (left panel), and one step of the proposed optimization strategy (right panel).
  • Figure 2: Learned state trajectories of HH model after training with RM, PR-SSM, NODE, NODE-LSTM, SPINODE methods and our proposed approach. Results are compared to ground truth ODE system trajectory labeled as GT. The proposed approach is capable of discerning the true trajectory for the unmeasured state $h_{gate}$.
  • Figure 3: The proposed approach's results for unknown initial conditions. Initial conditions $\hat{x}(t_{100})$ were learned using the first 100 samples.
  • Figure 4: Learned state trajectories of the cart-pole system after training RM, PR-SSM, SPINODE, NODE, NODE-LSTM methods and the proposed approach. Results are compared to ground truth ODE system trajectory labeled as GT. We showed that the proposed approach can discern the true trajectory for the unmeasured states $\dot{z}$ and $\dot{\phi}$.
  • Figure 5: Learned state trajectories of EMPS after training RM, PR-SSM, SPINODE, NODE, NODE-LSTM methods and the proposed approach. Results are compared to ground truth ODE system trajectory labeled as GT. The proposed approach can discern the true trajectory for the unmeasured state $\dot{q}_m$.
  • ...and 5 more figures

Theorems & Definitions (2)

  • Definition 1: State Identifiability
  • Theorem 1