One-sentence Summary

Drawing on classical nonlinear system identification, researchers from Robert Bosch GmbH and RPTU University Kaiserslautern-Landau propose a framework that exploits longer integration horizons during training of Physics-Enhanced Neural Ordinary Differential Equations to reduce bias in physical parameter estimates, extract residual information from data, and improve generalization, demonstrated by jointly estimating the flux map and resistance of a permanent magnet synchronous machine.

Key Contributions

• The paper establishes an explicit connection between Physics-Enhanced Neural ODE training and nonlinear system identification: explicit-step ODE solvers function as NARX predictors, and recurrent integration corresponds to a k-step-ahead NOE-like predictor.
• A training framework exploits longer integration horizons to reduce bias in physical parameter estimates, extract residual information from data, and act as a regularizer for improved generalization.
• Applied to a permanent magnet synchronous machine, the framework jointly estimates the flux linkage map and resistance, yielding reduced bias in physical parameter estimates and improved out-of-distribution prediction performance.

Introduction

Physics-Enhanced Neural ODEs (PeN-ODEs) integrate neural network components with explicit first-principle dynamical relations, offering data-efficient, physically interpretable models suitable for control tasks. During training, the integration horizon, the number of solver steps taken before computing the loss, is often treated as a hyperparameter, but its selection strongly influences training dynamics and estimator bias in ways that were not theoretically grounded. The authors establish a novel connection between Neural ODE training and classical nonlinear system identification, showing that explicit-step solvers behave as Nonlinear Autoregressive with Exogenous Input (NARX) predictors while recurrent integration over a horizon corresponds to a Nonlinear Output Error (NOE) simulation. This insight reveals that longer integration horizons reduce bias in physical parameter estimates by emphasizing long-term fidelity, at the cost of more challenging optimization. The framework is demonstrated on a real-world problem: identifying the flux linkage map and resistance of a Permanent Magnet Synchronous Machine from experimental data.

Method

The authors frame the training of Neural ODEs as a system identification problem, drawing a direct connection between the numerical integration step and a discrete‑time nonlinear predictor. Starting from a general ODE

they show that any explicit single‑step ODE solver with a fixed step size can be expressed as a one‑step‑ahead predictor

This formulation collects the system dynamics $f$f and the solver’s internal operations into a single function $g$g, which is essentially a Nonlinear Auto‑Regressive with eXogenous input (NARX) model. Multistep integrators are accommodated by allowing past values of the state and input, leading to the more general NARX form

The predictor representation is then extended to multi‑step integration. When the model is rolled out over a horizon $H$H starting from a measured initial condition $\boldsymbol{x}_0$x0​, the predicted states are generated recurrently:

with $\hat{\boldsymbol{x}}_0 = \boldsymbol{x}_0$x^0​=x0​. This structure is known as a $k$k‑step‑ahead predictor. If the initial condition is also taken from the model’s own predictions, the formulation becomes a Nonlinear Output‑Error (NOE) predictor. The training loss is the mean squared error over the complete horizon and a set of $N_{\mathrm{IC}}$NIC​ initial conditions drawn from the dataset:

In this framework, the training of a Neural ODE is directly interpreted as the identification of the one‑step‑ahead predictor $g$g, with the horizon $H$H controlling the trade‑off between NARX‑like ($H=1$H=1) and NOE‑like ($H>1$H>1) optimisation.

The selection of the integration horizon is guided by two key rationales. For models intended for long‑term simulation or open‑loop operation (e.g., model predictive control), a longer horizon reduces error accumulation and brings the behaviour closer to that of an output‑error model. For models that operate as part of a closed‑loop architecture, where the latest measurements are continuously fed back, one‑step‑ahead prediction suffices. A second, crucial motivation is bias minimisation. When the model structure cannot perfectly represent the true plant dynamics, training with $H=1$H=1 can yield biased and inconsistent parameter estimates. Increasing the horizon, despite making the optimisation non‑convex, helps mitigate this bias by forcing the model to predict accurately over a longer range, even in the presence of structural deficiencies.

The authors apply this general framework to the identification of a Permanent Magnet Synchronous Motor (PMSM). In the synchronously rotating $dq$dq‑frame, the voltage equations are

where the flux linkages are modelled as a parameterised nonlinear function $\boldsymbol{\psi}_{\mathrm{dq}} = \boldsymbol{\psi}(\boldsymbol{i}_{\mathrm{dq}}, \boldsymbol{\theta})$ψdq​=ψ(idq​,θ). Using the chain rule, the flux derivative is expressed as $\frac{\mathrm{d}}{\mathrm{d}t}\boldsymbol{\psi}_{\mathrm{dq}} = \boldsymbol{L}(\boldsymbol{i}_{\mathrm{dq}}, \boldsymbol{\theta}) \frac{\mathrm{d}}{\mathrm{d}t}\boldsymbol{i}_{\mathrm{dq}}$dtd​ψdq​=L(idq​,θ)dtd​idq​, with $\boldsymbol{L}$L the differential inductance matrix. Substituting this into the voltage equation yields a state‑space ODE for the currents:

This is a Physics‑enhanced Neural ODE (PeN‑ODE): the flux map $\boldsymbol{\psi}$ψ is a black‑box neural network, while the resistance $R$R is a physically interpretable parameter. Together with the chosen ODE solver, the integration step becomes a current predictor

where $\bar{\boldsymbol{u}}_k = [\boldsymbol{u}_{\mathrm{dq},k}, \omega_k]^T$uˉk​=[udq,k​,ωk​]T and $\bar{\boldsymbol{\theta}} = [R, \boldsymbol{\theta}]^T$θˉ=[R,θ]T. Training minimises the $k$k‑step‑ahead prediction error of the currents, allowing the model to learn the nonlinear flux map directly within the predictor structure. By choosing a horizon $H > 1$H>1, the framework can overcome the bias inherent in conventional one‑step flux estimation methods, producing a model that is both accurate and physically consistent.

Experiment

Experimental evaluation on simulation and test-bench data shows that increasing the prediction horizon during training reduces estimation bias from model-plant mismatch, yielding more physical resistance estimates and more accurate flux maps. Longer horizons improve generalization to unseen scenarios, acting as a regularizer that trades off a slight increase in training error. On real test-bench data with many unmodeled dynamics, the trend is non-monotonic but overall the technique helps extract information from imperfect data, while it offers no benefit for already well-designed identification experiments.