Dirac-Interconnected Neural Elements: Discovering Modularity in Physical Systems Without Reduction

Reiho Li; Razmik Arman Khosrovian; Takaharu Yaguchi; Hiroaki Yoshimura; Takashi Matsubara

cs.LGOct 2, 2026

Dirac-Interconnected Neural Elements: Discovering Modularity in Physical Systems Without Reduction

Authors: Reiho Li, Razmik Arman Khosrovian, Takaharu Yaguchi, Hiroaki Yoshimura, Takashi Matsubara

Organizations: Hokkaido University · Kyushu University · RIKEN · Waseda University

Abstract

Deep learning has shown remarkable success in the data-driven modeling of dynamical systems. Much of its success is attributed not to the flexibility of neural networks but to inductive biases based on physical prior knowledge, such as energy conservation and symplecticity. However, existing methods do not fully exploit the fact that real-world physical systems are interconnections of components. Some methods require the interconnection to be known a priori, while others assume the system to be reducible to an ordinary differential equation (ODE) and learn only the reduced ODE, discarding the algebraic constraints imposed by the interconnection. Here, we propose Dirac-interconnected neural elements (DINEs), a neural network model that represents a physical system as a differential-algebraic equation (DAE), whose algebraic constraints are given by a Dirac structure in kernel representation. With DINEs, we simultaneously identify from data the interconnection among the components as a Dirac structure and learn the characteristics of the components as neural networks. This allows us to keep the learned subsystems in unreduced form and isolate or compose them to make a new system without retraining. Moreover, DINEs can handle partially observable systems. Experimental results demonstrate these capabilities on physical systems beyond the reach of existing methods.

Figures & tables

Model	Ensuring Energetic Behavior	Identifying Interconnection	Learning Constraints	Learning from Partial Observation
Neural ODEs ( Chen et al., 2018 )	–	–	–	–
HNNs, LNNs, SymODENs, pHNNs †	✓	–	–	–
OE-pHNNs ( Moradi et al., 2026 )	✓	–	–	✓
CHNNs, CLNNs ( Finzi et al., 2020 )	✓	–	given §	–
Compositional pHNNs ( Neary & Topcu, 2023 )	✓	given	–	–
N-pHDAEs ( Neary et al., 2025 )	✓	given	given	–

Table 1: Comparison with related methods.

Figure 1: Conceptual diagram of DINEs.

Panel of Figure 2 — Figure 2: Diagrams of the systems of Tasks 1–5.

	(Task 1) Three-way junction
Model	MSE $\downarrow$	VPT $\uparrow$
Neural ODEs	$(2.871{\scriptstyle\pm 1.014})\times 10^{4}$	$0.062{\scriptstyle\pm 0.015}$
PoDiNNs ∗	$(6.129{\scriptstyle\pm 0.089})\times 10^{4}$	$0.000{\scriptstyle\pm 0.000}$
DINEs (soft) †	(not converged in 3 of 5 seeds)
DINEs (ours)	$\mathbf{3.319}{\scriptstyle\pm 1.221}$	$\mathbf{0.860}{\scriptstyle\pm 0.043}$
DINEs ( $n_{f}=1$ ) ‡	(not converged in all seeds)

Table 2: Experimental results.

Panel of Figure 3 — Figure 3: Visualizations of example predictions. Error denotes the absolute error.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

	Bond graph	Mechanical	Electromagnetic	Hydraulic
$f$	flow	velocity	current	volume flow
$e$	effort	force	voltage	pressure
$q$	time integral of flow	displacement	charge	volume
$p$	time integral of effort	momentum	flux linkage	flow tube momentum
C	C-type storage element	spring	capacitor	reservoir
I	I-type storage element	mass	inductor	fluid inertia

Appendix

Table 3: Power-conjugate variables, state variables, and elements in each domain

Task: system	C	I	R	Se	Sf	$n$	$(n_{f},n_{e})$	Observations	Inputs
1: Three-way junction	3	3	–	–	–	6	(2, 4)	6 states	–
2: RC ladder	4	–	7	2	–	13	(8, 5)	10 of 13 currents	2 source voltages
3: Subsystem A	3	3	2	1	–	9	(6, 3)	6 states	boundary force
3: Subsystem B	1	2	1	2	–	6	(4, 2)	3 states	boundary and external forces
4: Hydraulic	1	1	1	1	1	5	(3, 2)	2 states	source pressure, boundary flow
4: Mechanical	3	2	2	–	1	8	(5, 3)	5 states	boundary velocity

Appendix

Table 4: Systems of Tasks 1–4.

Resistor	$\Delta V_{j}$	$r_{j}$	$k_{j}$
$\mathrm{R}_{0}$	$u_{\mathrm{A}}-a$	0.8299	0.7991
$\mathrm{R}_{1}$	$a-v_{1}$	0.6711	4.6312
$\mathrm{R}_{2}$	$a-v_{2}$	0.7407	6.3965
$\mathrm{R}_{3}$	$b-v_{3}$	0.6981	6.5022
$\mathrm{R}_{4}$	$b-v_{4}$	0.5908	5.1852
$\mathrm{R}_{5}$	$a-b$	0.7813	2.3884

Appendix

Table 5: Resistors of Task 2.

Subsystem	Inertias	Springs $(a,b)$	Dampers $(a,b)$	Boundary (prescribed $\to$ recorded)	Observed
Task 3, A	$1,1.2,1.4$	$(0.4,0.1)$ $(0.3,0.1)$ $(0.2,0.1)$	$(0.07,0.45)$ $(0.09,0.55)$	$F_{\mathrm{b}}\to v$	$3\,q$ , $3\,v$
Task 3, B	$0.9,1.1$	$(0.35,0.15)$	$(0.08,0.5)$	$F_{\mathrm{b}}\to v$	$1\,q$ , $2\,v$
Task 4, hydraulic	$0.9$	$(0.8,0)$ (reservoir)	fluid resistor	$Q_{\mathrm{b}}\to P$	$1\,q$ , $1\,Q$
Task 4, mechanical	$1,1.4$	$(0.4,0.1)$ $(0.3,0.1)$ $(0.2,0.1)$	$(0.07,0.45)$ $(0.09,0.55)$	$v\to F_{\mathrm{b}}$	$3\,q$ , $2\,v$

Appendix

Table 6: Subsystems of Tasks 3 and 4.

	DINEs	Neural ODEs	PoDiNNs	CT-SUBNET	Coupled Neural ODEs
Training observations and known inputs	yes	yes	yes	yes	yes
Types and numbers of elements	yes	no	C-type, I-type	no	dimensions only
Correspondence of observations to elements	yes	as states	yes	as outputs	states, boundary port
Linearity of elements (coefficients unknown)	yes	no	I-type elements	no	no
$(n_{f},n_{e})$	yes	no	no	no	no
True wiring and coefficients	no	no	no	no	no

Appendix

Table 7: Prior knowledge given to the models. The last row is used for scoring only.

Task	Model	$\varepsilon=10^{-6}$	$10^{-5}$	$10^{-4}$	$10^{-3}$
1	Neural ODEs	$0.008\pm 0.001$	$0.023\pm 0.004$	$0.062\pm 0.015$	$0.160\pm 0.046$
1	PoDiNNs	$0.000\pm 0.000$	$0.000\pm 0.000$	$0.000\pm 0.000$	$0.001\pm 0.000$
1	DINEs (soft) †	$0.270\pm 0.160$	$0.548\pm 0.102$	$0.898\pm 0.022$	$1.000\pm 0.000$
1	DINEs (ours)	$0.204\pm 0.028$	$0.525\pm 0.031$	$0.860\pm 0.043$	$0.988\pm 0.011$
2	CT-SUBNET ( $n_{z}=4$ )	$0.000\pm 0.000$	$0.000\pm 0.000$	$0.039\pm 0.011$	$0.681\pm 0.036$
2	CT-SUBNET ( $n_{z}=8$ )	$0.000\pm 0.000$	$0.001\pm 0.000$	$0.096\pm 0.084$	$0.773\pm 0.037$

Appendix

Table 8: VPT of Tasks 1–5 for further thresholds

\varepsilon

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