Learning Reduced-Order Dynamics with Singularity via Latent-Augmented Neural Ordinary Differential Equations

Xiaorui Wang; Yu Zhou; Wenjie Mei; Dongzhe Zheng; Yang Bai; Masaaki Nagahara

cs.LGAug 22, 2026

Learning Reduced-Order Dynamics with Singularity via Latent-Augmented Neural Ordinary Differential Equations

Authors: Xiaorui Wang, Yu Zhou, Wenjie Mei, Dongzhe Zheng, Yang Bai, Masaaki Nagahara

Organizations: Lanzhou University of Technology, School of Automation and Electrical Engineering, Lanzhou, 730050, Gansu, China · Hiroshima University, Graduate School of Advanced Science and Engineering, Higashihiroshima, 739-0046, Hiroshima, Japan · Nanjing University - Suzhou Campus, School of Robotics and Automation, Suzhou, 215163, Jiangsu, China · Princeton University, Department of Mechanical and Aerospace Engineering, Princeton, 08544, New Jersey, United States

Abstract

This paper addresses the issue of self-intersecting trajectories (in phase space) in industrial reduced-order modeling and proposes the Latent-Augmented Neural Ordinary Differential Equations (LA-NODEs) framework. From the perspective of artificial intelligence, the proposed method augments conventional neural ordinary differential equations to enhance model expressiveness, enabling the representation of conflicting vector fields that may arise in reduced-order systems, thereby improving learning accuracy. Through theoretical analysis, the underlying mechanism of the framework is established, and a condition for determining the minimum required augmentation dimension is derived. From the perspective of engineering applications, the effectiveness of the proposed method is validated on the reduced-order system of two representative industrial models, namely an interior permanent magnet synchronous motor (IPMSM) drive and a distributed energy system (DES). Experimental results demonstrate that the proposed method can recover system features that are difficult to capture using conventional approaches and achieve superior performance in terms of prediction accuracy and modeling fidelity, thereby providing an effective approach for high-precision data-driven modeling of complex industrial systems.

Figures & tables

Figure 1: Comparison of learned and true trajectories for example system.

Figure 2: Architecture of the LA-NODE model, including primary neural dynamics (Block 1, Eq. ( 8a )), latent neural dynamics (Block 2, Eq. ( 8b )), adjoint primary neural dynamics (Block 3, Eq. ( 11a )), and adjoint latent neural dynamics (Block 4, Eq. ( 11b )). $a_{y_{t}}$ and $a_{z_{t}}$ represent the adjoint state defined as Eq. ( 10 ). Blue lines denote forward propagation, red lines denote adjoint (loss-driven) backward propagation, and green lines denote parameter updates as shown in Eq. ( 12 ).

Parameter	Symbol	Value [Unit]
Rated power	$P$	150 [W]
Rated speed	$N_{r}$	1000 [rpm]
Rated load	$T_{L}$	1.2 [N $\cdot$ m]
Rated voltage	$U_{r}$	24 [V]
Rated current	$I_{r}$	7.5 [A]
Number of pole pairs	$n_{p}$	4 [–]

Table 1: Key parameters of the IPMSM system

Parameter	Symbol	Value [Unit]
Maximum battery capacity	$E_{\text{bat,max}}$	5000 [kWh]
Charge/discharge efficiency	$\eta_{\text{bat}}$	0.95
Inertia constant	$M$	10 [s]
Damping coefficient	$D$	1 [p.u.]
Rated angular frequency	$\omega_{0}$	$100\pi$ [rad/s]
PV dynamic time constant	$T_{\text{PV}}$	5 [s]

Table 2: Key parameters of the DES

Figure 3: Comparison of trajectory-wise errors (

Error=\Big\|\sqrt{(i_{d,true}(i)-i_{d,pred}(i))^{2}+(i_{q,true}(i)-i_{q,pred}(i))^{2}}\Big\|

) across four frameworks over the full dataset for IPMSM. Index represents the data point position in the IPMSM dataset.

Panel of Figure 4 — Figure 4: Performance comparison between CSODE and LA-NODE based on IPMSM trajectories: (a) Phase portraits, (b) Time-domain responses.

Panel of Figure 5 — Figure 5: Comparison of MSE Loss across different frameworks with epochs for IPMSM: (a) Training set, (b) Testing set.

Panel of Figure 6 — Figure 6: Comparison of state variables for DES (the color variation indicates the progression of time): (a) LA-NODE, (b) ANODE, (c) CSODE, (d) Transformer, for $x_{3}$ ; (e) LA-NODE, (f) ANODE, (g) CSODE, (h) Transformer, for $x_{4}$ .

Model	MAE	RMSE	Max Error	Std Error
LA-NODE (ours)	0.0075 $\pm$ 0.0037	0.0097 $\pm$ 0.0049	0.0270 $\pm$ 0.0138	0.0061 $\pm$ 0.0033
CSODE	0.1771 $\pm$ 0.0053	0.2266 $\pm$ 0.0046	0.5819 $\pm$ 0.0095	0.1404 $\pm$ 0.0008
Transformer	0.2625 $\pm$ 0.0404	0.3517 $\pm$ 0.0422	0.8586 $\pm$ 0.0498	0.2333 $\pm$ 0.0240
ANODE	0.0171 $\pm$ 0.0085	0.0236 $\pm$ 0.0114	0.0858 $\pm$ 0.0484	0.0161 $\pm$ 0.0083

Table 3: Comparison of four dynamic modeling frameworks for DES

Figure 7: Comparison of MSE Loss for the testing set across four frameworks for DES.

Figure 8: Comparison of MSE Loss (on logarithmic scale) for training set and testing set across different augmentation dimensions.

$k$	1	2	3	4	5
Time (sec)	637.2	638.2	648.9	684.0	740.2

Table 4: Average total training time (sec) over 5 runs for 700 epochs, varying

k

Figure 9: Trade-off analysis between computational cost and test loss under different augmentation dimensions.

Reduced-Order Modeling

Reduced-order modeling builds compact approximations of complex systems to make simulation, prediction, and control more efficient.

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Université Paris-Saclay, CEA List, Palaiseau, France

Lagrangian Neural Networks Reduced-Order Modeling Latent Dynamics Modeling Nonlinear System Identification

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This work proposes a latent Lagrangian-based framework for reduced-order modelling of forced nonlinear dynamical systems. In contrast with conventional Lagrangian or Hamiltonian neural networks, our approach learns a set of latent coordinates sufficient to capture the dynamics conjointly with two neural networks for the latent kinetic and latent potential energies, and leverages force supervision to eliminate the need for an ODE solver during training. Consistency of physical laws in the latent space is ensured through the principle of virtual work. Results show that the model effectively learns the subtle dynamics induced by the system's nonlinearity and non-convex potential energy, and generalizes well to unseen forces and initial conditions. These observations confirm the physical relevance of the proposed approach, and its interest for model reduction.

Abstract

Reduced-Order Modeling

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