Buckingham–Symplectic Networks (BuSyNet)

• Buckingham–Symplectic Networks are a deep learning framework that learns Hamiltonians by mapping phase-space trajectories to latent action–angle variables.
• The architecture uniquely integrates symplectic structure with dimensional consistency using a Buckingham-π inspired symbolic head for precise, unit-aware Hamiltonian formulation.
• Empirical results on canonical systems like the harmonic oscillator and Kepler problems show superior long-term accuracy, stability, and interpretability compared to standard baselines.

Searching arXiv for the BuSyNet paper and closely related Hamiltonian-learning work. Buckingham–Symplectic Networks (BuSyNet) are a deep learning architecture for Hamiltonian discovery that combines two inductive biases usually treated separately: dimensional consistency and symplectic action–angle structure. In BuSyNet, a strictly symplectic encoder maps observed phase-space trajectories $(q,p)$ to latent action–angle variables $(I,\theta)$, and a Buckingham-$\pi$–inspired symbolic head learns a Hamiltonian $H(I,m)$ with units of energy by constraining monomial exponents through $D_{\text{in}}\Psi = D_{\text{out}}$ (Germany et al., 1 Apr 2026). The method is formulated for integrable Hamiltonian systems, where canonical transformations to action–angle coordinates reduce the dynamics to $\dot I = 0$ and $\dot \theta = \partial H/\partial I$, and it is evaluated on the harmonic oscillator and the Kepler two-body problem in two and three dimensions, where it recovers concise closed-form Hamiltonians and yields lower long-horizon state error and smaller energy variance than NN, HNN, and SympNet baselines (Germany et al., 1 Apr 2026).

1. Concept and scope

BuSyNet is motivated by the observation that the Hamiltonian is not merely a scalar predictor but a physical quantity with units of energy and a generator of symplectic dynamics. The framework therefore treats dimensional homogeneity and canonical structure as coupled constraints rather than independent regularizers. The first constraint is that the Hamiltonian must carry units $M L^2 T^{-2}$; the second is that, for integrable systems, there exists a symplectic change of variables to action–angle coordinates $(I,\theta)$ in which trajectories become rigid rotations on invariant tori (Germany et al., 1 Apr 2026).

The architecture is described as learning a symplectic encoder $S_\phi$ from observed phase-space trajectories to latent action–angle variables and a symbolic Hamiltonian head that combines the learned actions with known physical parameters. This design is intended to identify Hamiltonians precisely rather than only up to an additive constant, because the training procedure uses precomputed action integrals from the data as anchors for the latent variables (Germany et al., 1 Apr 2026).

Within the cited work, BuSyNet targets integrable Hamiltonian systems with full phase-space observations and known units for the relevant variables and parameters. The harmonic oscillator and the Kepler problem serve as canonical examples because both admit action–angle representations and closed-form Hamiltonians in those coordinates (Germany et al., 1 Apr 2026).

2. Mathematical framework

BuSyNet is built on the standard Hamiltonian formulation. For $(I,\theta)$0 and scalar Hamiltonian $(I,\theta)$1, Hamilton’s equations are

$(I,\theta)$2

or, equivalently,

$(I,\theta)$3

with

$(I,\theta)$4

where $(I,\theta)$5 (Germany et al., 1 Apr 2026).

The relevant geometric structure is symplecticity. A differentiable map $(I,\theta)$6 is symplectic if its Jacobian satisfies

$(I,\theta)$7

for all $(I,\theta)$8. Canonical transformations preserve the symplectic form, and compositions of symplectic maps remain symplectic (Germany et al., 1 Apr 2026). BuSyNet uses this property to ensure that the learned encoder is not merely approximately structure-preserving but symplectic by construction.

The latent representation is based on action–angle variables. For integrable Hamiltonians there exists a canonical transformation to $(I,\theta)$9 such that $\pi$0 and

$\pi$1

In these coordinates, motion lies on invariant $\pi$2-tori and becomes a rigid rotation with constant frequency vector $\pi$3 (Germany et al., 1 Apr 2026). This latent-space form is central to BuSyNet: the learned dynamics are constrained so that actions remain constant and angles evolve linearly.

3. Dimensional analysis and symbolic Hamiltonian construction

The unit-aware component of BuSyNet is a Buckingham-$\pi$4–inspired symbolic head based on dimensional matrices and monomial parameterization. Each input variable $\pi$5 is assigned a base-units vector $\pi$6, and these are collected into

$\pi$7

A general monomial

$\pi$8

has units vector $\pi$9 (Germany et al., 1 Apr 2026). Classical Buckingham analysis obtains dimensionless groups from the nullspace equation $H(I,m)$0; BuSyNet extends this idea to produce outputs with prescribed units.

In the symbolic head, for a mini-batch $H(I,m)$1,

$H(I,m)$2

where $H(I,m)$3 contains trainable exponents. The unit constraint is imposed softly by

$H(I,m)$4

with loss

$H(I,m)$5

For Hamiltonian discovery, $H(I,m)$6 is the energy units vector $H(I,m)$7 (Germany et al., 1 Apr 2026).

The 1D oscillator illustrates the constraint. A candidate monomial $H(I,m)$8 must satisfy

$H(I,m)$9

that is,

$D_{\text{in}}\Psi = D_{\text{out}}$0

The symbolic head learns $D_{\text{in}}\Psi = D_{\text{out}}$1 subject to the dimensional constraint (Germany et al., 1 Apr 2026). This mechanism narrows the admissible hypothesis class to forms with correct physical units, which the source identifies as important for extrapolation across parameter scales and interpretability.

The symbolic discovery proceeds in two stages. First, the head learns a latent monomial of the form

$D_{\text{in}}\Psi = D_{\text{out}}$2

with exponents trained under the unit constraint. Second, after $D_{\text{in}}\Psi = D_{\text{out}}$3 is identified and rounded to $D_{\text{in}}\Psi = D_{\text{out}}$4, the model replaces the intermediate form with an action-only expression. If $D_{\text{in}}\Psi = D_{\text{out}}$5,

$D_{\text{in}}\Psi = D_{\text{out}}$6

If $D_{\text{in}}\Psi = D_{\text{out}}$7,

$D_{\text{in}}\Psi = D_{\text{out}}$8

For one-degree-of-freedom systems such as simple harmonic motion, the BuckiNet monomial itself already matches the true form and is used directly (Germany et al., 1 Apr 2026).

4. Architecture and training procedure

The encoder in BuSyNet is a G-SympNet layer stack $D_{\text{in}}\Psi = D_{\text{out}}$9 that is symplectic by construction. The building blocks are canonical shear maps:

$\dot I = 0$0

$\dot I = 0$1

with learnable functions $\dot I = 0$2 and $\dot I = 0$3 implemented as small neural networks. Alternating compositions such as $\dot I = 0$4 remain symplectic, and each block is exactly invertible:

$\dot I = 0$5

The reported implementation uses a 4-layer G-SympNet with 32 neurons per hidden layer (Germany et al., 1 Apr 2026).

A distinctive component is the use of precomputed action integrals to anchor the latent actions. For each action,

$\dot I = 0$6

where the period $\dot I = 0$7 is estimated by FFT and the integral is evaluated numerically with Simpson’s rule (Germany et al., 1 Apr 2026). These numerically estimated actions are then matched to the learned $\dot I = 0$8 through a reconstruction term, which the source presents as crucial for identifying $\dot I = 0$9 precisely.

Training uses three principal losses:

$\dot \theta = \partial H/\partial I$0

$\dot \theta = \partial H/\partial I$1

$\dot \theta = \partial H/\partial I$2

Since $\dot \theta = \partial H/\partial I$3 depends only on $\dot \theta = \partial H/\partial I$4, one has $\dot \theta = \partial H/\partial I$5, so the ideal latent dynamics satisfy $\dot \theta = \partial H/\partial I$6 (Germany et al., 1 Apr 2026). Time derivatives are computed by the chain rule,

$\dot \theta = \partial H/\partial I$7

using automatic differentiation through $\dot \theta = \partial H/\partial I$8, with $\dot \theta = \partial H/\partial I$9 and $M L^2 T^{-2}$0 obtained by numerically differentiating the trajectories (Germany et al., 1 Apr 2026).

The pipeline is four-stage. First, trajectories $M L^2 T^{-2}$1 and known physical parameters are collected, and $M L^2 T^{-2}$2 and $M L^2 T^{-2}$3 are formed. Second, actions are precomputed by FFT-based period estimation and Simpson’s-rule integration. Third, the symbolic head is pretrained on $M L^2 T^{-2}$4 by minimizing $M L^2 T^{-2}$5 with L-BFGS for 100 epochs and rounding learned exponents to the nearest $M L^2 T^{-2}$6. Fourth, the G-SympNet encoder and Hamiltonian coefficients are optimized with Adam for 1000 epochs at learning rate $M L^2 T^{-2}$7 using

$M L^2 T^{-2}$8

with $M L^2 T^{-2}$9 in all experiments (Germany et al., 1 Apr 2026).

At inference time, latent-space rollout proceeds by encoding $(I,\theta)$0 to $(I,\theta)$1, computing $(I,\theta)$2, updating the angles with an Euler step,

$(I,\theta)$3

keeping actions fixed,

$(I,\theta)$4

and decoding via $(I,\theta)$5 to obtain $(I,\theta)$6 (Germany et al., 1 Apr 2026). Because the symplectic blocks are exactly invertible, decoding is performed exactly by reversing the G-SympNet blocks.

5. Empirical results and recovered Hamiltonians

The reported experiments cover the harmonic oscillator and the Kepler two-body problem in 2D and 3D. For the harmonic oscillator,

$(I,\theta)$7

with action–angle form

$(I,\theta)$8

The setup uses $(I,\theta)$9, $S_\phi$0, $S_\phi$1, and $S_\phi$2, with 1000 training points from $S_\phi$3 and testing on the remainder (Germany et al., 1 Apr 2026).

For the Kepler problem, the Hamiltonian is given in spherical coordinates as

$S_\phi$4

with $S_\phi$5. In action–angle variables the 3D Hamiltonian is

$S_\phi$6

and the 2D version omits $S_\phi$7 (Germany et al., 1 Apr 2026). The 3D setup uses $S_\phi$8, $S_\phi$9, $(I,\theta)$00, $(I,\theta)$01, and initial conditions $(I,\theta)$02, $(I,\theta)$03, $(I,\theta)$04, $(I,\theta)$05, $(I,\theta)$06, $(I,\theta)$07, corresponding to a circular orbit of radius $(I,\theta)$08; training uses $(I,\theta)$09, testing uses $(I,\theta)$10, and the period is $(I,\theta)$11. The 2D setup uses $(I,\theta)$12 with the same train-test split and $(I,\theta)$13 (Germany et al., 1 Apr 2026).

The discovered symbolic Hamiltonians are reported as follows.

System	Recovered Hamiltonian
Simple harmonic oscillator	$(I,\theta)$14
Kepler 2D	$(I,\theta)$15
Kepler 3D	$(I,\theta)$16

These expressions are stated to be consistent with the analytic forms for the chosen parameter settings, and the exponents are reported to match analytic values to rounding precision $(I,\theta)$17 (Germany et al., 1 Apr 2026). For the oscillator, the recovered exponents $(I,\theta)$18 on $(I,\theta)$19 and $(I,\theta)$20 on $(I,\theta)$21 are explicitly identified as physically meaningful in the source.

The recovered frequencies are also listed. For SHM, $(I,\theta)$22 for $(I,\theta)$23. For the circular-orbit Kepler cases, the reported values are $(I,\theta)$24 and $(I,\theta)$25 in 2D, and $(I,\theta)$26, $(I,\theta)$27, and $(I,\theta)$28 in 3D (Germany et al., 1 Apr 2026). This is consistent with the latent-space interpretation in which constant actions and linearly advancing angles directly expose the frequencies $(I,\theta)$29.

6. Accuracy, stability, interpretability, and limitations

The reported quantitative evaluation emphasizes long-horizon accuracy and stability. For the harmonic oscillator with period $(I,\theta)$30, the mean squared error on $(I,\theta)$31 is reported as NN $(I,\theta)$32, HNN $(I,\theta)$33, SympNet $(I,\theta)$34, and BuSyNet $(I,\theta)$35; on $(I,\theta)$36 it is NN $(I,\theta)$37, HNN $(I,\theta)$38, SympNet $(I,\theta)$39, and BuSyNet $(I,\theta)$40 (Germany et al., 1 Apr 2026). The energy variance $(I,\theta)$41 is NN $(I,\theta)$42, HNN $(I,\theta)$43, SympNet $(I,\theta)$44, and BuSyNet $(I,\theta)$45, which the source characterizes as best by 5–6 orders of magnitude (Germany et al., 1 Apr 2026).

For Kepler 2D over 25 periods, the state MSE is NN $(I,\theta)$46, HNN $(I,\theta)$47, SympNet $(I,\theta)$48, and BuSyNet $(I,\theta)$49; the energy variance is NN $(I,\theta)$50, HNN $(I,\theta)$51, SympNet $(I,\theta)$52, and BuSyNet $(I,\theta)$53 (Germany et al., 1 Apr 2026). For Kepler 3D over 25 periods, the MSE is NN $(I,\theta)$54, HNN $(I,\theta)$55, SympNet $(I,\theta)$56, and BuSyNet $(I,\theta)$57; the energy variance is NN $(I,\theta)$58, HNN $(I,\theta)$59, SympNet $(I,\theta)$60, and BuSyNet $(I,\theta)$61 (Germany et al., 1 Apr 2026).

The source does not report explicit ablations, but it states that comparisons to HNN and SympNet act as implicit ablations. Relative to HNN, adding the action–angle latent space plus units is said to reduce drift and phase error substantially. Relative to SympNet, adding the unit-aware symbolic Hamiltonian yields more accurate long-term prediction and orders-of-magnitude smaller energy variance (Germany et al., 1 Apr 2026). Symbolic sparsity penalties and alternative latent dimensions are explicitly noted as unexplored.

Interpretability in BuSyNet has two components. First, the symbolic Hamiltonian makes physical content explicit. The oscillator example,

$(I,\theta)$62

recovers the textbook action–angle form $(I,\theta)$63 and corresponds to the standard Hamiltonian $(I,\theta)$64 in canonical variables (Germany et al., 1 Apr 2026). Second, the latent variables themselves are interpretable: actions are constant, angles advance linearly, and frequencies are directly available from $(I,\theta)$65 (Germany et al., 1 Apr 2026).

The limitations stated in the source are substantial and delineate the scope of the method. BuSyNet assumes integrability and global action–angle coordinates; non-integrable or chaotic systems are not targeted. It assumes full $(I,\theta)$66 observations and known base-unit assignments for inputs and outputs. Parameters entering the Hamiltonian must be present among the inputs to be discoverable. Numerical derivatives and action-integral estimates may be noise-sensitive, so smoothing or robust differentiation may be necessary. Increasing degrees of freedom enlarges the multi-index set $(I,\theta)$67 and raises SympNet capacity requirements, although end-to-end training is described as feasible for low-to-moderate dimensions (Germany et al., 1 Apr 2026). A plausible implication is that the method’s strongest regime is low-dimensional, cleanly observed, integrable mechanics with explicit physical parameters.

7. Relation to prior work and terminological clarifications

The BuSyNet paper situates itself against two strands of prior Hamiltonian learning. Hamiltonian Neural Networks learn a Hamiltonian and derive dynamics via Hamilton’s equations, but they do not enforce units and do not reduce dynamics to action–angle coordinates. Symplectic networks such as SympNet enforce symplecticity, but they are typically trained as time-steppers and do not combine a learned canonical transformation with a dimensionally constrained symbolic Hamiltonian (Germany et al., 1 Apr 2026). Within that comparison, BuSyNet is presented as pairing a symplectic action–angle encoder with a dimensionally aware symbolic head that outputs a closed-form Hamiltonian in correct energy units.

A separate terminological clarification concerns possible confusion with an unrelated use of “Buckingham–Symplectic Networks.” The paper “A ‘network of networks’ (from history to algebra)” works with complex-valued flows and tensions, lattices in $(I,\theta)$68, Siegel space, and symplectic group actions, and it explicitly states that it does not discuss Buckingham $(I,\theta)$69 dimensional analysis (Parrochia, 2023). Its “connection to Buckingham–Symplectic Networks” is framed as a path for integrating Buckingham $(I,\theta)$70 into a lattice-and-Siegel framework rather than as the BuSyNet architecture for Hamiltonian discovery (Parrochia, 2023). This suggests that the term may be used in two distinct contexts: one in Hamiltonian system identification from trajectories (Germany et al., 1 Apr 2026), and another in an algebraic-geometric construction over network lattices and Siegel modular forms (Parrochia, 2023).

In the Hamiltonian-learning sense, BuSyNet denotes the architecture that integrates a symplectic encoder, precomputed action integrals, and a Buckingham-$(I,\theta)$71–inspired symbolic Hamiltonian head. Its defining claim is that jointly enforcing symplectic structure and dimensional consistency yields stable, accurate, and interpretable Hamiltonian discovery for integrable systems (Germany et al., 1 Apr 2026).