Discovery of Symbolic Hamiltonian Expressions with Buckingham-Symplectic Networks

The paper introduces BuSyNet, a deep learning architecture that integrates dimensional consistency and symplectic geometry to discover interpretable, closed-form symbolic Hamiltonian expressions, achieving superior long-term prediction accuracy and stability on physical systems like the harmonic oscillator and Kepler problem compared to state-of-the-art methods.

The Gist

Imagine you are trying to predict the future path of a planet, a swinging pendulum, or a bouncing ball. In the world of physics, these objects follow strict rules: they don't just lose energy randomly, and they move in specific, repeating patterns.

For a long time, scientists have used Deep Learning (a type of super-smart computer brain) to try to predict these movements. But standard AI often makes mistakes. It might predict the planet will slowly spiral into the sun because it forgot the rule that energy must be conserved, or it might get the math units wrong (like mixing up meters and seconds).

This paper introduces a new, smarter AI called BuSyNet (Buckingham-Symplectic Network). Think of BuSyNet not just as a calculator, but as a physics detective that solves two major mysteries at once:

1. The "Shape" of the Motion: It understands the hidden geometry of how things move.
2. The "Units" of the Motion: It ensures the math makes sense physically (e.g., you can't add apples to oranges).

Here is how it works, broken down with simple analogies:

1. The Problem: The "Drifting" AI

Imagine you are watching a child on a swing. If you ask a normal AI to predict where the swing will be in 10 minutes, it might say, "The swing will stop," or "The swing will go higher and higher." Why? Because the AI is just guessing patterns from the data without understanding the rules of the swing. Over time, these small guesses add up, and the prediction drifts wildly off course.

2. The Solution: The "Magic Map" (Symplectic Geometry)

The authors realized that for systems like swings or planets, there is a special "magic map" (called Action-Angle coordinates).

• The Old Way: Trying to predict the swing's exact position and speed at every moment is like trying to describe a circle by listing every single dot on its edge. It's messy and prone to error.
• The BuSyNet Way: BuSyNet uses a special layer (a Symplectic Network) to translate the messy "position and speed" data into a simpler language: Action and Angle.
  • Action: Think of this as the "size" of the swing. Once the swing starts, the size of its path never changes. It's a constant.
  • Angle: Think of this as the "position on the clock." It just ticks forward at a steady, predictable rate.

By translating the data into this "Action-Angle" language, the AI sees the motion as a straight, boring line instead of a complex curve. This makes predicting the future incredibly easy and accurate because the "Action" never changes, and the "Angle" just moves in a straight line.

3. The Second Problem: The "Unit" Confusion

Even if the AI gets the shape right, it might still get the scale wrong.

• The Analogy: Imagine a recipe that says "Add 5 cups of flour and 2 grams of sugar." If you mix them, the cake is ruined. In physics, you can't mix Mass, Time, and Length randomly. The final answer (Energy) must have specific "units" (like Joules).
• The BuSyNet Fix: The authors added a special layer called BuckiNet (based on the Buckingham-π theorem). This layer acts like a strict librarian. It checks every math operation to ensure the units match.
  • If the AI tries to calculate energy, the librarian checks: "Does this formula result in units of Energy? If not, I will adjust the math until it does."
  • This forces the AI to discover the actual formula (like $E = \frac{1}{2}mv^2$) rather than just a random curve that fits the data.

4. The Result: The "Perfect Recipe"

When you combine the Magic Map (Symplectic) and the Strict Librarian (BuckiNet), you get BuSyNet.

The authors tested this on two classic problems:

1. The Harmonic Oscillator: A simple spring bouncing up and down.
2. The Kepler Problem: A planet orbiting a star (like Earth around the Sun).

The Outcome:

• Accuracy: While other AI models started to drift and lose accuracy after a few hours of simulation, BuSyNet stayed perfect for days. It didn't lose energy; it respected the laws of physics.
• Interpretability: Most AI models are "black boxes"—you put data in, and a number comes out, but you don't know why. BuSyNet is different. It actually spits out the mathematical formula for the energy. It doesn't just predict the planet's path; it tells you the exact equation that governs the universe for that system.

Summary

Think of BuSyNet as a translator that speaks two languages fluently:

1. The Language of Geometry: It knows that the universe has a hidden, simple structure (the invariant torus) and uses that to predict the future without drifting.
2. The Language of Units: It knows that physics has strict rules about dimensions (Mass, Length, Time) and refuses to make a calculation unless the units make sense.

By combining these two, BuSyNet doesn't just guess the future; it discovers the laws of physics hidden inside the data, giving us a tool that is accurate, stable, and actually understandable by humans.

Technical Summary

1. Problem Statement

Hamiltonian systems are fundamental to modeling physical dynamics, characterized by energy conservation and symplectic geometry. While recent deep learning approaches (e.g., Hamiltonian Neural Networks, SympNets) have successfully modeled these systems by embedding physical constraints into architectures or loss functions, they suffer from two critical limitations:

1. Lack of Dimensional Consistency: Existing models often mix variables with heterogeneous physical units (e.g., mass, length, time) without enforcing dimensional consistency. This limits interpretability and the ability to extrapolate to new physical regimes.
2. Ignored Canonical Transformations: Most methods do not explicitly leverage the fact that integrable Hamiltonian systems admit a canonical transformation to action-angle coordinates. In these coordinates, the dynamics simplify to uniform rotation on an invariant torus, where actions are constant and angles evolve linearly.

The authors aim to bridge this gap by developing a framework that simultaneously discovers symbolic, dimensionally consistent Hamiltonian expressions and learns the symplectic transformation to action-angle variables.

2. Methodology: BuSyNet Architecture

The authors propose BuSyNet (Buckingham-Symplectic Networks), a two-stage deep learning architecture that integrates symplectic geometry with dimensional analysis.

A. Core Components

1. Symplectic Encoder (SympNet):

   • Inspired by SympNets, this component learns a strictly symplectic, invertible map $\mathcal{S}_\phi: (q, p) \to (I, \theta)$.
   • It transforms observed position-momentum trajectories into latent action-angle variables $(I, \theta)$.
   • In this latent space, the dynamics are simplified: actions $I$ are constant, and angles $\theta$ evolve linearly ($\dot{\theta} = \text{const}$).

2. Dimensional Consistency Head (BuckiNet):

   • Inspired by the Buckingham-$\pi$ theorem and the BuckiNet architecture, this layer ensures the output Hamiltonian has correct physical units (Energy).
   • It takes the latent actions $I$ and known system parameters $m$ (e.g., mass, spring constant) as input.
   • It learns a symbolic expression of the form $\hat{H} = \sum a_k \prod I_j^{\alpha_j} \prod m_i^{\beta_i}$, where the exponents are constrained such that the resulting expression is dimensionally consistent (i.e., $D_{in}\Psi = D_{out}$, where $D_{out}$ represents Energy units).

B. Training Strategy (Three-Stage Process)

The training is designed to satisfy three specific aims:

1. Grounding Action Calculations: Before training the neural network, the authors numerically compute the "ground truth" action values $I$ from the trajectory data using the integral definition $I = \frac{1}{2\pi} \oint p \, dq$. This provides a target for the encoder.
2. Dimensional Pre-training: The BuckiNet head is trained first (using L-BFGS) to learn the correct exponents for the system parameters and actions to satisfy the dimensional constraint ($L_{dim}$). This yields the functional form of the Hamiltonian.
3. Joint Optimization: The SympNet encoder and the Hamiltonian coefficients are trained jointly using Adam. The total loss function combines:
   • Action Reconstruction Loss ($L_I$): Minimizes the difference between the learned actions and the pre-computed numerical actions.
   • Equation of Motion Loss ($L_{\dot{\theta}}, L_{\dot{I}}$): Enforces Hamilton's equations in the latent space ($\dot{\theta} = \partial H / \partial I$ and $\dot{I} = -\partial H / \partial \theta$).

C. Inference and Forecasting

Once trained, BuSyNet performs long-term forecasting by:

1. Mapping input $(q, p)$ to $(I, \theta)$ via the encoder.
2. Advancing the angle $\theta$ linearly in time (since $I$ is constant).
3. Mapping the updated $(I, \theta)$ back to $(q, p)$ via the inverse encoder.
   This process avoids error accumulation typical in direct time-stepping.

3. Key Contributions

• BuSyNet Architecture: The first deep learning framework to combine symplectic geometry (for stability) with dimensional analysis (for physical consistency) to discover symbolic Hamiltonians.
• Symbolic Discovery: Unlike standard neural networks that output black-box functions, BuSyNet recovers closed-form, analytical expressions for the Hamiltonian with correct physical units.
• Grounded Action Learning: By pre-computing action integrals from data, the method resolves the ambiguity often found in learning action-angle variables, ensuring the learned transformation is unique and physically meaningful.
• Superior Long-Term Stability: The architecture guarantees that the learned dynamics evolve on an invariant torus, eliminating phase drift and energy drift over long time horizons.

4. Experimental Results

The authors evaluated BuSyNet on two canonical integrable systems: the 1D Harmonic Oscillator and the Kepler Two-Body Problem (2D and 3D).

• Symbolic Recovery:
  • Harmonic Oscillator: Recovered $\hat{H}(I) \approx I \sqrt{k/m}$, matching the analytic form $H = \omega I$.
  • Kepler Problem: Recovered $\hat{H}(I) \approx -mk^2 / (2I^2)$, matching the analytic form for both 2D and 3D cases.
• Prediction Accuracy:
  • BuSyNet significantly outperformed Vanilla Neural Networks, Hamiltonian Neural Networks (HNN), and SympNets in Mean Squared Error (MSE) for long-term rollouts (up to $20T$ and $25T$).
  • Energy Conservation: BuSyNet maintained energy variance ($Var(\hat{H})$) on the order of $10^{-11}$ to $10^{-12}$, whereas baselines showed variances ranging from $10^{-3}$ to $10^{3}$ (drifting significantly).
• Phase Locking: Trajectories predicted by BuSyNet remained phase-locked with the ground truth, while baselines diverged after a few oscillation periods.

5. Significance and Conclusion

This work demonstrates that embedding dimensional analysis and symplectic structure into deep learning pipelines is not merely an aesthetic choice but a practical necessity for scientific machine learning.

• Interpretability: The model outputs human-readable equations rather than opaque weights.
• Generalization: By respecting physical units, the model generalizes better across different parameter scales and regimes.
• Stability: The latent torus structure ensures that long-term predictions do not suffer from the numerical instability common in standard ODE solvers or neural networks.

The authors conclude that BuSyNet provides a robust pathway for discovering physically faithful models from data, though future work is needed to extend these methods to non-integrable (chaotic) systems and scenarios with partial observations (e.g., position-only data).