cs.LGFeb 17, 2026

Controlled oscillation modeling using port-Hamiltonian neural networks

Authors: Maximino Linares, Guillaume Doras, Thomas Hélie

Organizations: IRCAM, 1, place Igor-Stravinsky, 75004, Paris, France

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Abstract

Learning dynamical systems through purely data-driven methods is challenging as they do not learn the underlying conservation laws that enable them to correctly generalize. Existing port-Hamiltonian neural network methods have recently been successfully applied for modeling mechanical systems. However, even though these methods are designed on power-balance principles, they usually do not consider power-preserving discretizations and often rely on Runge-Kutta numerical methods. In this work, we propose to use a second-order discrete gradient method embedded in the learning of dynamical systems with port-Hamiltonian neural networks. Numerical results are provided for three systems deliberately selected to span different ranges of dynamical behavior under control: a baseline harmonic oscillator with quadratic energy storage; a Duffing oscillator, with a non-quadratic Hamiltonian offering amplitude-dependent effects; and a self-sustained oscillator, which can stabilize in a controlled limit cycle through the incorporation of a nonlinear dissipation. We show how the use of this discrete gradient method outperforms the performance of a Runge-Kutta method of the same order. Experiments are also carried out to compare two theoretically equivalent port-Hamiltonian systems formulations and to analyze the impact of regularizing the Jacobian of port-Hamiltonian neural networks during training.

Figures & tables

SystemHarmonic oscillatorDuffing oscillatorSelf-sustained oscillator
 (linear)(nonlinear)(nonlinear)
\begin{array}[]{l}\bm{x}=\begin{bmatrix}q\\ p\end{bmatrix}\\ \bm{u}\end{array}\begin{array}[]{rll}q:&\text{centered position (elongation)}&\text{[m]}\\ p:&\text{momentum}&\text{[Kg.m.s{}^{-1}]}\\ u=f:&\text{force (applied or exterior)}&\text{[N]}\end{array}
H(x)H(\bm{x})p22m+kq22\displaystyle\frac{p^{2}}{2m}+\frac{kq^{2}}{2}p22m+k1q22+k3q44\displaystyle\frac{p^{2}}{2m}+\frac{k_{1}q^{2}}{2}+\frac{k_{3}q^{4}}{4}p22m+kq22\displaystyle\frac{p^{2}}{2m}+\frac{kq^{2}}{2}
J\bm{J}\bm{J}=\left[\begin{array}[]{cc|c}0&1&0\\ -1&0&-1\\ \hline\cr 0&1&0\end{array}\right]\bm{J}=\left[\begin{array}[]{cc|c}0&-1&0\\ 1&0&0\\ \hline\cr 0&0&0\end{array}\right]
R\bm{R}\bm{R}=\alpha\,\left[\begin{array}[]{cc|c}0&0&0\\ 0&1&0\\ \hline\cr 0&0&0\end{array}\right] with α>0\alpha>0 [N /ms−1/ms^{-1} ]\bm{R}=\Gamma(w)\left[\begin{array}[]{cc|c}1&0&1\\ 0&0&0\\ \hline\cr 1&0&1\end{array}\right] with Γ(w)>0\Gamma(w)>0 [ m/sm/s N]
S\bm{S}\bm{S}=\left[\begin{array}[]{cc|c|c}0&1&0&0\\ -1&0&-1&-1\\ \hline\cr 0&1&0&0\\ 0&1&0&0\end{array}\right]\bm{S}=\left[\begin{array}[]{cc|c|c}0&-1&1&0\\ 1&0&0&0\\ \hline\cr-1&0&0&-1\\ 0&0&1&0\end{array}\right]
Table 1: Different port-Hamiltonian structural elements, Jacobian matrices and related quantities for the considered oscillators.
Figure 1: Architecture of the two PHNN models considered in this work. White boxes with orange contour denote fixed algebraic operations whereas orange boxes indicate the trainable parameters.
 (i) [x˙y]=(J−R(x,u))[∇H(x)u]\begin{bmatrix}\bm{\dot{x}}\\ \bm{y}\end{bmatrix}=(\bm{J}-\bm{R}(\bm{x},\bm{u}))\begin{bmatrix}\nabla H(\bm{x})\\ \bm{u}\end{bmatrix}(ii) [x˙wy]=S[∇H(x)z(w)u]\begin{bmatrix}\bm{\dot{x}}\\ \bm{w}\\ \bm{y}\end{bmatrix}=\bm{S}\begin{bmatrix}\nabla H(\bm{x})\\ \bm{z}(\bm{w})\\ \bm{u}\end{bmatrix}
H(x)H(\bm{x})HθH(x)=12xTLθLH(x)TLθLH(x)xH_{\theta_{H}}(\bm{x})=\frac{1}{2}\bm{x}^{T}\bm{L}_{\theta_{L_{H}}}(\bm{x})^{T}\bm{L}_{\theta_{L_{H}}}(\bm{x})\bm{x}
DissipationRθR(w)=LθLR(x,u)TLθLR(x,u)\bm{R}_{\theta_{R}}(\bm{w})=\bm{L}_{\theta_{L_{R}}}(\bm{x},\bm{u})^{T}\bm{L}_{\theta_{L_{R}}}(\bm{x},\bm{u})zθz(w)=ΓθΓ(w)wΓθΓ(w)=(LθLzT(w)LθLz(w))+(KθKz(w)−KθKzT(w))\begin{aligned} \bm{z}_{\theta_{z}}(\bm{w})&=\bm{\Gamma}_{\theta_{\Gamma}}(\bm{w})\bm{w}\\ \bm{\Gamma}_{\theta_{\Gamma}}(\bm{w})&=(\bm{L}_{\theta_{L_{z}}}^{T}(\bm{w})\bm{L}_{\theta_{L_{z}}}(\bm{w}))\\ &+(\bm{K}_{\theta_{K_{z}}}(\bm{w})-\bm{K}_{\theta_{K_{z}}}^{T}(\bm{w}))\end{aligned}
Interconnection matrixJ∈R(nx+nu)×(nx+nu)\bm{J}\in\mathbb{R}^{(n_{x}+n_{u})\times(n_{x}+n_{u})}S∈R(nx+nw+nu)×(nx+nw+nu)\bm{S}\in\mathbb{R}^{(n_{x}+n_{w}+n_{u})\times(n_{x}+n_{w}+n_{u})}
Table 2: Fixed and learned objects for the different port-Hamiltonian networks based on formulations (i) and (ii).
Figure 2: Training and inference diagram for the continuous models fθ\bm{f}_{\theta} .
Figure 3: Training and inference diagram for the discrete models gθ\bm{g}_{\theta} .
Accuracy orderNameCharacteristicPower-balance
2Explicit midpoint (RK2)Explicit×\times
2Discrete gradient (DG)Inverse-explicit✓\checkmark
Table 3: Accuracy order, name, characteristic and whether power-balance is respected for the chosen numerical schema.
SystemHarmonic OscillatorDuffing OscillatorSelf-sustained oscillator
Intrinsic parameters(m,k)(m,k)(m,k1,k3)(m,k_{1},k_{3})(m,k)(m,k)
f0f_{0} [Hz]1Hz1Hz
mm [kg]0.160.16
1/k,1/k11/k,1/k_{1} [ N/mN/m ]0.160.16
k3k_{3} [ N/m3N/m^{3} ]-100k1100k_{1}-
z(w)z(w)0.9w0.9w [N]1.3w3−4w2+3w1.3w^{3}-4w^{2}+3w [m/s][m/s]
Table 4: Implementation hyperparameters for the different experiments.
Figure 4: Training points, test initial points and two complete test trajectories for each of the three oscillatory systems. Note that in the case of the harmonic and Duffing oscillator, the applied control shifted the equilibrium point from (p,q)=(0,0)(p,q)=(0,0) whereas in the case of the self-sustained oscillator, it stabilizes the trajectories in a limit cycle.
Figure 5: Schematic sampling and dataset construction procedure. A trajectory generated at sampling frequency srgen{sr}_{gen} over a duration D=Dinfer=βT0D=D_{infer}=\beta T_{0} is shown as white dot markers. From this trajectory, a training point, highlighted with a green star marker, is uniformly sampled from a subset of samples, shown as black dot markers, obtained at frequency srtrain{sr}_{train} and restricted to t≤αT0t\leq\alpha T_{0} . The training horizon DtrainD_{train} and the inference horizon DinferD_{infer} are indicated by arrows, with the vertical dashed line marking the end of the training interval.
Figure 6: Boxplot of the inference errors for the three different oscillators and varying numbers of training points (small models). From left to right: harmonic, Duffing and self-sustained oscillators.
Figure 7: Comparison of the learned trajectory discretizations for each oscillatory system starting on x0\bm{x}_{0} such that H(x0)=0.5JH(\bm{x}_{0})=0.5J . ( Left ) NODE and PHNN-S models trained on Ntrain=25N_{train}=25 from the harmonic oscillator. ( Center ) NODE and PHNN-S models trained on Ntrain=100N_{train}=100 from the Duffing oscillator. ( Right ) NODE and PHNN-JR models trained on Ntrain=25N_{train}=25 from the self-sustained oscillator. For the harmonic and Duffing oscillator, the trajectory is obtained with u=0\bm{u}=0 ; and for the self-sustained, with u\bm{u} such that the system stabilizes in a limit cycle.
Figure 8: Boxplot of the inference errors for the three different oscillators and model sizes ( Ntrain=25N_{train}=25 ). From left to right: harmonic, Duffing and self-sustained oscillators.
Figure 9: Comparison of the learned trajectory discretizations by the discrete gradient for each oscillatory system starting on x0\bm{x}_{0} such that H(x0)=0.75JH(\bm{x}_{0})=0.75J . ( Left ) PHNN-S models trained on Ntrain=25N_{train}=25 from the harmonic oscillator. ( Center ) PHNN-S models trained on Ntrain=25N_{train}=25 from the Duffing oscillator. ( Right ) PHNN-JR models trained on Ntrain=25N_{train}=25 from the self-sustained oscillator. For the harmonic and Duffing oscillator, the trajectory is obtained with u=0\bm{u}=0 ; and for the self-sustained, with u\bm{u} such that the system stabilizes in a limit cycle.
Figure 10: Mean condition number ( left ), spectral norm ( center ) and stiffness ratio ( right ) per optimizer step during the training for the different architectures when modeling the Duffing oscillator with Ntrain=25N_{train}=25 points and no Jacobian regularization (BL). For each optimizer step, the corresponding mean value (solid line) is obtained computing the average over the 10 model initializations, with the shaded area indicating the range between the minimum and maximum values.
Figure 11: Boxplot of the inference errors for the three different oscillators and normalizations ( Ntrain=25N_{train}=25 and small number of parameters). Notation: BL refers to the baseline (i.e. no Jacobian regularization); CN, to the condition number regularization ( 34 ); SN, to the spectral norm regularization ( 35 ); and SR, to the stiffness ratio regularization ( 36 ). From left to right: harmonic, Duffing and self-sustained oscillators.
Figure 12: Impact on the mean condition number ( left ), spectral norm ( center ) and stiffness ratio ( right ) per optimizer step during the training for the different architectures when modeling the Duffing oscillator with Ntrain=25N_{train}=25 points and the proposed Jacobian regularizations. For each optimizer step, the corresponding mean value (solid line) is obtained computing the average over the 10 model initializations, with the shaded area indicating the range between the minimum and maximum values.
Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

SystemHarmonic oscillator (linear)
Jf(x)\bm{J}_{\bm{f}}(\bm{x})[01m−k−αm]\begin{bmatrix}0&\frac{1}{m}\\ -k&-\frac{\alpha}{m}\end{bmatrix}
∥Jf(x)∥2\|\bm{J}_{\bm{f}}(\bm{x})\|_{2}12(k2+1+α2m2+(k2+1+α2m2)2−4k2m2)\sqrt{\frac{1}{2}\left(k^{2}+\frac{1+\alpha^{2}}{m^{2}}+\sqrt{\left(k^{2}+\frac{1+\alpha^{2}}{m^{2}}\right)^{2}-\frac{4k^{2}}{m^{2}}}\right)}
κ(Jf(x))\kappa(\bm{J}_{\bm{f}}(\bm{x}))k2+1+α2m2+(k2−1+α2m2)2+4k2α2m2k2+1+α2m2−(k2−1+α2m2)2+4k2α2m2\sqrt{\frac{k^{2}+\tfrac{1+\alpha^{2}}{m^{2}}+\sqrt{\left(k^{2}-\tfrac{1+\alpha^{2}}{m^{2}}\right)^{2}+\tfrac{4k^{2}\alpha^{2}}{m^{2}}}}{k^{2}+\tfrac{1+\alpha^{2}}{m^{2}}-\sqrt{\left(k^{2}-\tfrac{1+\alpha^{2}}{m^{2}}\right)^{2}+\tfrac{4k^{2}\alpha^{2}}{m^{2}}}}}
ρ(Jf(x))\rho(\bm{J}_{\bm{f}}(\bm{x}))1, if α2<4km (underdamped)1,~if~\alpha^{2}<4km~(\text{underdamped})
Appendix
Table 5: Jacobian matrix, spectral norm, condition number and stiffness ratio for the harmonic oscillator.
SystemDuffing oscillator (nonlinear)
Jf(x)\bm{J}_{\bm{f}}(\bm{x})[01m−k1−3k3q2−αm]\begin{bmatrix}0&\frac{1}{m}\\ -k_{1}-3k_{3}q^{2}&-\frac{\alpha}{m}\end{bmatrix}
∥Jf(x)∥2\|\bm{J}_{\bm{f}}(\bm{x})\|_{2}12((k1+3k3q2)2+1+α2m2+((k1+3k3q2)2+1+α2m2)2−4(k1+3k3q2)2m2)\sqrt{\frac{1}{2}\left((k_{1}+3k_{3}q^{2})^{2}+\frac{1+\alpha^{2}}{m^{2}}+\sqrt{\left((k_{1}+3k_{3}q^{2})^{2}+\frac{1+\alpha^{2}}{m^{2}}\right)^{2}-\frac{4(k_{1}+3k_{3}q^{2})^{2}}{m^{2}}}\right)}
κ(Jf(x))\kappa(\bm{J}_{\bm{f}}(\bm{x}))(k1+3k3q2)2+1+α2m2+((k1+3k3q2)2−1+α2m2)2+4α2(k1+3k3q2)2m2(k1+3k3q2)2+1+α2m2−((k1+3k3q2)2−1+α2m2)2+4α2(k1+3k3q2)2m2\sqrt{\frac{(k_{1}+3k_{3}q^{2})^{2}+\tfrac{1+\alpha^{2}}{m^{2}}+\sqrt{\left((k_{1}+3k_{3}q^{2})^{2}-\tfrac{1+\alpha^{2}}{m^{2}}\right)^{2}+\tfrac{4\alpha^{2}(k_{1}+3k_{3}q^{2})^{2}}{m^{2}}}}{(k_{1}+3k_{3}q^{2})^{2}+\tfrac{1+\alpha^{2}}{m^{2}}-\sqrt{\left((k_{1}+3k_{3}q^{2})^{2}-\tfrac{1+\alpha^{2}}{m^{2}}\right)^{2}+\tfrac{4\alpha^{2}(k_{1}+3k_{3}q^{2})^{2}}{m^{2}}}}}
ρ(Jf(x))\rho(\bm{J}_{\bm{f}}(\bm{x}))1, if α2<4m(k1+3k3q2)\alpha^{2}<4m(k_{1}+3k_{3}q^{2}) ( underdamped )
Appendix
Table 6: Jacobian matrix, spectral norm, condition number and stiffness ratio for the Duffing oscillator.
SystemSelf-sustained oscillator (nonlinear)
Jf(x)\bm{J}_{\bm{f}}(\bm{x})[−Γ′(w)kq−Γ(w)k−1mk0]\begin{bmatrix}-\Gamma^{\prime}(w)kq-\Gamma(w)k&-\frac{1}{m}\\ k&0\end{bmatrix}
∥Jf(x)∥2\|\bm{J}_{\bm{f}}(\bm{x})\|_{2}12(k2(Γ′(w)q+Γ(w))2+k2+1m2+(k2(Γ′(w)q+Γ(w))2+k2+1m2)2−4k2m2)\sqrt{\frac{1}{2}\left(k^{2}\big(\Gamma^{\prime}(w)q+\Gamma(w)\big)^{2}+k^{2}+\frac{1}{m^{2}}+\sqrt{\left(k^{2}\big(\Gamma^{\prime}(w)q+\Gamma(w)\big)^{2}+k^{2}+\frac{1}{m^{2}}\right)^{2}-\frac{4k^{2}}{m^{2}}}\right)}
κ(Jf(x))\kappa(\bm{J}_{\bm{f}}(\bm{x}))k2(Γ′(w)q+Γ(w))2+k2+1m2+(k2(Γ′(w)q+Γ(w))2+k2−1m2)2+4k2(Γ′(w)q+Γ(w))2m2k2(Γ′(w)q+Γ(w))2+k2+1m2−(k2(Γ′(w)q+Γ(w))2+k2−1m2)2+4k2(Γ′(w)q+Γ(w))2m2\sqrt{\frac{k^{2}(\Gamma^{\prime}(w)q+\Gamma(w))^{2}+k^{2}+\tfrac{1}{m^{2}}+\sqrt{\left(k^{2}(\Gamma^{\prime}(w)q+\Gamma(w))^{2}+k^{2}-\tfrac{1}{m^{2}}\right)^{2}+\tfrac{4k^{2}(\Gamma^{\prime}(w)q+\Gamma(w))^{2}}{m^{2}}}}{k^{2}(\Gamma^{\prime}(w)q+\Gamma(w))^{2}+k^{2}+\tfrac{1}{m^{2}}-\sqrt{\left(k^{2}(\Gamma^{\prime}(w)q+\Gamma(w))^{2}+k^{2}-\tfrac{1}{m^{2}}\right)^{2}+\tfrac{4k^{2}(\Gamma^{\prime}(w)q+\Gamma(w))^{2}}{m^{2}}}}}
ρ(Jf(x))\rho(\bm{J}_{\bm{f}}(\bm{x}))1, if k2(Γ′(w)q+Γ(w))2<4kmk^{2}(\Gamma^{\prime}(w)q+\Gamma(w))^{2}<\frac{4k}{m} ( underdamped )
Appendix
Table 7: Jacobian matrix, spectral norm, condition number and stiffness ratio for the self-sustained oscillator.
 q←2E0k1sin⁡θ,p←2mE0cos⁡θq\leftarrow\sqrt{\frac{2E_{0}}{k_{1}}}\sin\theta,\qquad p\leftarrow\sqrt{2mE_{0}}\cos\theta 
Appendix
Algorithm 1 Initial condition sampling for HO and SSO
ModelComponentsNetworkInputLayersHiddenOutputNumber of parameters
  typedimension unitsdimension(per component)(per model)
NODEBlack-boxMLP33100221,4k21,4k
PHNN-JRLθLH(x)\mathbf{L}_{\theta_{L_{H}}}(\mathbf{x})MLP22100310,7k21,8k
LθLR(x,u)\mathbf{L}_{\theta_{L_{R}}}(\mathbf{x},\mathbf{u})MLP32100311,1k
PHNN-SLθLH(x)\mathbf{L}_{\theta_{L_{H}}}(\mathbf{x})MLP22100310,7k21,1k
LθLz(w)\mathbf{L}_{\theta_{L_{z}}}(\mathbf{w})MLP12100110,4k
Appendix
Table 8: Specificities of the implemented port-Hamiltonian neural networks with large number of parameters.
ModelComponentsNetworkInputLayersHiddenOutputNumber of parameters
  typedimension unitsdimension(per component)(per model)
NODEBlack-boxMLP326024,3k4,3k
PHNN-JRLθLH(x)\mathbf{L}_{\theta_{L_{H}}}(\mathbf{x})MLP224232,1k4,3k
LθLR(x,u)\mathbf{L}_{\theta_{L_{R}}}(\mathbf{x},\mathbf{u})MLP324232,2k
PHNN-SLθLH(x)\mathbf{L}_{\theta_{L_{H}}}(\mathbf{x})MLP224232,1k4,1k
LθLz(w)\mathbf{L}_{\theta_{L_{z}}}(\mathbf{w})MLP124212k
Appendix
Table 9: Specificities of the implemented port-Hamiltonian neural networks with medium size of parameters.
ModelComponentsNetworkInputLayersHiddenOutputNumber of parameters
  typedimension unitsdimension(per component)(per model)
NODEBlack-boxMLP32242848848
PHNN-JRLθLH(x)\mathbf{L}_{\theta_{L_{H}}}(\mathbf{x})MLP22163371809
LθLR(x,u)\mathbf{L}_{\theta_{L_{R}}}(\mathbf{x},\mathbf{u})MLP32163438
PHNN-SLθLH(x)\mathbf{L}_{\theta_{L_{H}}}(\mathbf{x})MLP22163371852
LθLz(w)\mathbf{L}_{\theta_{L_{z}}}(\mathbf{w})MLP12201481
Appendix
Table 10: Specificities of the implemented port-Hamiltonian neural networks with low size of parameters.
Harmonic25100400
NODE-RK22.53e-02 [2.33e-02]1.69e-04 [1.54e-04]2.79e-05 [1.84e-05]
PHNN-JR-DG2.58e-04 [3.82e-04]9.10e-06 [1.22e-05]7.29e-06 [5.07e-06]
PHNN-JR-RK21.14e-04 [1.02e-04]6.88e-06 [3.81e-06]5.18e-06 [5.50e-06]
PHNN-S-DG4.22e-05 [5.99e-05]6.24e-06 [4.97e-06]4.78e-06 [1.21e-05]
PHNN-S-RK24.89e-05 [2.11e-04]5.77e-06 [9.55e-06]3.73e-06 [8.89e-06]
Duffing25100400
Appendix
Table 11: Median inference error and IQR (in brackets) for different numbers of training points when learning the three oscillatory systems with small models. In bold, the best result for each combination of system and number of training points.
HarmonicSmall sizeMedium sizeLarge size
NODE-RK22.53e-02 [2.33e-02]2.85e-02 [2.59e-02]1.45e-01 [4.67e-02]
PHNN-JR-DG2.58e-04 [3.82e-04]1.47e-04 [2.68e-04]7.33e-04 [2.52e-03]
PHNN-JR-RK21.14e-04 [1.02e-04]5.56e-05 [1.19e-04]1.51e-04 [3.87e-04]
PHNN-S-DG4.22e-05 [5.99e-05]5.56e-05 [9.58e-05]2.45e-04 [4.59e-04]
PHNN-S-RK24.89e-05 [2.11e-04]4.28e-05 [3.73e-05]2.01e-04 [2.13e-04]
DuffingSmall sizeMedium sizeLarge size
Appendix
Table 12: Median inference error and IQR (in brackets) for different model sizes when learning the three oscillatory systems with Ntrain=25N_{train}=25 . In bold, the best result for each combination of system and model size.
HarmonicBLCNSNSR
NODE-RK22.53e-02 [2.33e-02]7.82e-02 [1.63e+00]2.63e-02 [2.53e-02]2.56e-02 [5.70e-02]
PHNN-JR-DG2.58e-04 [3.82e-04]2.20e-04 [3.40e-04]1.94e-04 [3.09e-04]2.30e-04 [3.19e-04]
PHNN-JR-RK21.14e-04 [1.02e-04]1.50e-04 [5.70e-04]7.18e-05 [9.39e-05]1.26e-04 [1.46e-04]
PHNN-S-DG4.22e-05 [5.99e-05]3.84e-05 [4.05e-05]3.44e-05 [6.62e-05]1.13e-04 [3.07e-04]
PHNN-S-RK24.89e-05 [2.11e-04]1.17e-05 [5.67e-05]5.66e-05 [2.02e-04]2.97e-05 [5.35e-04]
DuffingBLCNSNSR
Appendix
Table 13: Median inference error and IQR (in brackets) after applying the different Jacobian regularizations when learning the three oscillatory systems with Ntrain=25N_{train}=25 and small models. The best result for each combination of system and Jacobian regularization is shown in bold. Notation: BL refers to the baseline (i.e. no Jacobian regularization); CN, to the condition number regularization ( 34 ); SN, to the spectral norm regularization ( 35 ); and SR, to the stiffness ratio regularization ( 36 )

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