Research map / Probabilistic and scientific modeling / Differential equations and PDEs
Learning Physics Informed Neural ODEs with Partial Measurements
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Proposes a sequential, alternating Newton-based (Kalman-like) recursive framework for learning neural ODE dynamics of unmeasured states under partial measurements, and reports lower nRMSE than RM, PR-SSM, SPINODE, NODE and NODE-LSTM on five simulated systems and one real positioning dataset.
Problem
Learning ODE dynamics when some states are unmeasured and the dynamics generating those unmeasured states are unknown; joint state-parameter estimation is ill-posed and gradients vanish under partial measurement.
Approach
Sequentially optimizes latent states and then model parameters at each time step using alternating Newton updates that yield Kalman-filter-like recursions, so parameters are learned from estimated (not measured) unmeasured states, with a physics-informed loss on the known dynamics.
Key results
- Proposed approach achieves nRMSE of 1.54e-1 on HH vs 2.39e-1 for RM (best baseline) and 9.41e-3 on cart-pole vs 2.84e-1 for NODE.
- On the real EMPS dataset, the proposed method reaches nRMSE 9.50e-2 versus 6.20e-1 for RM and 1.65 for NODE.
- Harmonic oscillator nRMSE 5.08e-3 for proposed vs 2.36e-2 for RM, a modest advantage over RM.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use the proposed sequential alternating Newton approach instead of RM | Partially measured dynamics learning · HH model | nRMSE: 2.39 \cdot 10^{-1} -> 1.54 \!\cdot\! 10^{-1} | Helped |
| use the proposed sequential alternating Newton approach instead of SPINODE | Partially measured dynamics learning · Yeast Glyco. | nRMSE: 4.98 \cdot 10^{-2} -> 3.39 \!\cdot\! 10^{-2} | Helped |
| use the proposed sequential alternating Newton approach instead of NODE | Partially measured dynamics learning · Cart-pole | nRMSE: 2.84 \cdot 10^{-1} -> 9.41\!\cdot\! 10^{-3} | Helped |
| use the proposed sequential alternating Newton approach instead of RM | Partially measured dynamics learning · Harmonic Osc. | nRMSE: 2.36 \cdot 10^{-2} -> 5.08\!\cdot\! 10^{-3} | Helped |
| use the proposed sequential alternating Newton approach instead of RM | Partially measured dynamics learning · EMPS | nRMSE: 6.20 \cdot 10^{-1} -> 9.50 \!\cdot\! 10^{-2} | Helped |
Surprising findings
- Joint estimation of states and parameters suffers vanishing gradients under partial measurements, since the Kalman gain for parameters vanishes when the unmeasured state's dynamics are the only parameter-dependent term.
Limitations
- Assumes state identifiability, which is hard to guarantee for neural network dynamics and when unmeasured states outnumber measurements.
- Baselines (GP-based and other Bayesian methods) were not benchmarked because code was unavailable or they were computationally infeasible.
- Experiments use Euler integration and simulated systems with known ground-truth; only one real dataset.
Experimental setup
Datasets: Hodgkin-Huxley simulated data cart-pole simulated data (LQR controlled) yeast glycolysis simulated data harmonic oscillator simulated data retinal circulation simulated data electro-mechanical positioning system real dataset
Benchmarks: Hodgkin-Huxley neuron model Yeast glycolysis model Cart-pole system Harmonic oscillator Electro-mechanical positioning system Retinal circulation model
Metrics: nRMSE
Baselines: RM PR-SSM SPINODE NODE NODE-LSTM
Builds on: Neural ODE Physics-informed neural ODE Kalman filter SPINODE PR-SSM
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