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

$$\dot{\boldsymbol{x}} = f(\boldsymbol{x}, \boldsymbol{u}, \boldsymbol{\theta})$$x˙=f(x,u,θ)

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

$$\hat{\boldsymbol{x}}_{k+1} = g(\boldsymbol{x}_k, \boldsymbol{u}_k, \boldsymbol{\theta}).$$x^k+1​=g(xk​,uk​,θ).

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

$$\hat{\boldsymbol{x}}_{k+1} = g(\boldsymbol{x}_k, \dots, \boldsymbol{x}_{k-n_x}, \boldsymbol{u}_k, \dots, \boldsymbol{u}_{k-n_u}, \boldsymbol{\theta}).$$x^k+1​=g(xk​,…,xk−nx​​,uk​,…,uk−nu​​,θ).

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:

$$\hat{\boldsymbol{x}}_{k+1} = g(\hat{\boldsymbol{x}}_k, \boldsymbol{u}_k, \boldsymbol{\theta}) \quad \text{for } k = 0,\dots, H-1,$$x^k+1​=g(x^k​,uk​,θ)for k=0,…,H−1,

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:

$$L(\boldsymbol{\theta}) = \frac{1}{H N_{\mathrm{IC}}} \sum_{j=1}^{N_{\mathrm{IC}}} \sum_{k=0}^{H-1} \big\| \boldsymbol{x}_{k+1}^{(j)} - g(\hat{\boldsymbol{x}}_k^{(j)}, \boldsymbol{u}_k, \boldsymbol{\theta}) \big\|^2.$$L(θ)=HNIC​1​j=1∑NIC​​k=0∑H−1​​xk+1(j)​−g(x^k(j)​,uk​,θ)​2.

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

$$\boldsymbol{u}_{\mathrm{dq}} = R \boldsymbol{i}_{\mathrm{dq}} + \omega J \boldsymbol{\psi}_{\mathrm{dq}} + \frac{\mathrm{d}}{\mathrm{d}t}\boldsymbol{\psi}_{\mathrm{dq}},$$udq​=Ridq​+ωJψdq​+dtd​ψdq​,

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:

$$\frac{\mathrm{d}}{\mathrm{d}t}\boldsymbol{i}_{\mathrm{dq}} = \boldsymbol{L}^{-1}(\boldsymbol{i}_{\mathrm{dq}}, \boldsymbol{\theta}) \big[ -R \boldsymbol{i}_{\mathrm{dq}} - \omega J \boldsymbol{\psi}(\boldsymbol{i}_{\mathrm{dq}}, \boldsymbol{\theta}) + \boldsymbol{u}_{\mathrm{dq}} \big].$$dtd​idq​=L−1(idq​,θ)[−Ridq​−ωJψ(idq​,θ)+udq​].

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

$$\hat{\boldsymbol{i}}_{\mathrm{dq}, k+1} = g\big(\boldsymbol{i}_{\mathrm{dq}, k}, \bar{\boldsymbol{u}}_k, \bar{\boldsymbol{\theta}}\big),$$i^dq,k+1​=g(idq,k​,uˉk​,θˉ),

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.