🤖 machine learning

Learning partially observed systems with neural Hamiltonian ordinary differential equations

This paper introduces Neural Hamiltonian Ordinary Differential Equations (NHODE), a framework that combines Hamiltonian neural networks with neural ODEs to learn partially observed dynamical systems by enforcing energy conservation and physical constraints, thereby achieving superior accuracy and long-term stability compared to purely data-driven baselines.

Original authors: Sunniva Meltzer, Sølve Eidnes, Alexander Johannes Stasik

Published 2026-05-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Sunniva Meltzer, Sølve Eidnes, Alexander Johannes Stasik

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to teach a robot to predict how a complex machine moves. Usually, you'd show the robot a video of the whole machine and say, "Watch this, learn the rules, and then tell me what happens next."

But what if the machine has a secret compartment? You can see the gears on the outside turning, but you can't see the gears hidden inside. The outside gears are moving because of the hidden ones, but you have no camera inside to see them. If you try to teach a standard robot (a "pure data" learner) just by showing it the outside, it will eventually get confused. It will guess wrong about the hidden gears, and soon its prediction of the whole machine will fall apart, spinning wildly off course.

This paper introduces a new kind of robot teacher called NHODE (Neural Hamiltonian Ordinary Differential Equations). It's designed specifically to solve this "blind spot" problem by teaching the robot to respect the laws of physics rather than just memorizing patterns.

Here is how it works, using simple analogies:

1. The Problem: The "Blind" Learner

In the real world, we often can't measure everything. Maybe we can see a car's speed but not the fuel level, or we can see a planet's position but not the invisible gravity pulling it.

The Old Way (Pure Data): Imagine a student trying to learn a dance by only watching the dancer's feet. If the dancer's upper body does something unexpected, the student gets lost because they don't understand the connection between the feet and the rest of the body. In the paper, these "pure data" models failed quickly, especially in chaotic systems (like the "Three-Body Problem," which is like three planets dancing in a wild, unpredictable waltz).

2. The Solution: The "Physics-First" Teacher

The NHODE framework is like a teacher who says, "Don't just watch the feet; remember that energy cannot be created or destroyed."

The Energy Bank (Hamiltonian): The paper uses a concept called a "Hamiltonian," which is essentially a bank account for the system's total energy. In a closed system, this energy balance must always stay the same.
The Trick: The NHODE model is built so that it cannot break this rule. It forces the math to obey the conservation of energy. Even if the robot can't see the hidden gears, it knows that if the visible gears speed up, the hidden ones must be slowing down somewhere else to keep the total energy balance correct.

3. How It Handles the "Hidden" Parts

The paper describes a clever two-step process:

The Detective (Encoder): First, the model looks at the short history of the visible parts (the "observed history"). It acts like a detective, using the visible clues to guess what the hidden parts are doing right now. It infers the "initial state" of the invisible gears.
The Simulator (Rollout): Once it has a guess for the hidden parts, it runs a simulation forward in time. Crucially, it only checks its homework against the parts it can actually see. It doesn't get punished for guessing the hidden parts wrong directly; it only gets punished if the visible parts don't match the real video.
- Analogy: Imagine playing a game of "Simon Says" where you can only see Simon's hands. You have to guess what his feet are doing. If your guess about the feet causes your prediction of the hands to be wrong, you lose. But if your guess about the feet keeps the hands moving correctly, you win.

4. Adding Extra Rules (Symmetry)

The authors didn't stop at just energy. They added more "common sense" rules:

Translation Symmetry: If you move the whole machine one foot to the left, the physics shouldn't change. The model learns to look at the distance between parts rather than their absolute position.
Separable Energy: They assumed the energy is split into "movement energy" (which they know the formula for) and "potential energy" (which the AI learns). This makes the job easier and more accurate.

5. The Results: Stability in Chaos

The paper tested this on three scenarios, from simple to very hard:

Two Masses on Springs: A simple system. All methods worked okay, but the physics-based one was slightly better.
Three Masses in a Triangle: A bit more complex. The pure data models started to drift and make big mistakes. The physics-based model stayed on track.
The Three-Body Problem: This is the "boss level." Three planets pulling on each other in a chaotic dance.
- The Outcome: The pure data models (the "blind" learners) went crazy and their predictions exploded into nonsense very quickly.
- The Winner: The NHODE model, especially the one with all the extra physics rules (conserving energy, momentum, and angular momentum), kept predicting the dance correctly for a very long time. It didn't just guess; it understood the rules of the dance, so even when it couldn't see one dancer, it knew how they had to move to keep the dance going.

Summary

The paper claims that by building physical laws directly into the AI's brain, we can teach it to understand systems even when we can't see all the parts. It turns a "blind guesser" into a "physics-savvy detective" that can infer the invisible from the visible, keeping its predictions stable even in chaotic, unpredictable environments.

The authors note that while the model can predict the visible parts perfectly, the exact state of the hidden parts might not be unique (there could be a few different hidden scenarios that look the same from the outside), but the model finds a version that is physically consistent and accurate for the parts we care about.

Technical Summary: Learning Partially Observed Systems with Neural Hamiltonian Ordinary Differential Equations

Problem Statement

The paper addresses the challenge of learning dynamical systems from data when the system state is only partially observed. In many physical scenarios, certain state variables are inaccessible to measurement, yet they influence the evolution of the observed variables. Standard physics-informed machine learning models, such as Hamiltonian Neural Networks (HNNs), typically require access to the full system state (all positions and momenta) and their time derivatives to define the loss function. This requirement renders them inapplicable to partially observed settings where latent variables must be inferred without direct supervision. Conversely, purely data-driven approaches, such as standard Neural Ordinary Differential Equations (Neural ODEs), often fail to generalize or maintain stability in these settings, particularly for complex or chaotic systems, because they lack the physical constraints necessary to constrain the solution space.

Methodology

The authors propose Neural Hamiltonian Ordinary Differential Equations (NHODE), a framework that integrates the structural guarantees of Hamiltonian mechanics with the flexible training procedures of Neural ODEs.

Core Framework

The system is modeled as a canonical Hamiltonian system:
$\dot{x} = S \nabla H(x)$
where $x = (q, p)^T$ represents generalized position and momentum, $H$ is the Hamiltonian (total energy), and $S$ is the canonical symplectic matrix.

The NHODE framework distinguishes itself through three key components:

Rollout-Based Training with Partial Loss: Unlike standard HNNs that compute loss based on the residual of Hamilton's equations at every time step (requiring full state data), NHODE adopts the Neural ODE training paradigm. The model integrates the learned vector field forward from an initial condition to generate a full trajectory. The loss function is defined only on the observed components of the state ( $x_{obs}$ ) at the data time points. The latent components ( $x_{hid}$ ) are excluded from the loss calculation, meaning their influence on training is indirect, mediated solely through the evolution of the observed variables.
Latent State Inference: To initialize the integration, the framework employs a neural network encoder to infer the initial state of the unobserved variables ( $x_{hid}(t_0)$ ) based on the initial history of observed variables. This encoder and the Hamiltonian network are trained jointly using a shared loss function.
Physical Constraints via Symmetry and Separability: To further constrain the solution space and improve generalization, the authors incorporate specific physical priors:
- Coordinate Transformations: For systems with translational symmetry, absolute coordinates are transformed into pairwise relative distances before being passed to the neural network. This enforces translation invariance, ensuring the conservation of linear momentum.
- Separable Hamiltonians: The framework supports models where the Hamiltonian is assumed separable ( $H = T(p) + V(q)$ ) with a known kinetic energy term ( $T(p) = p^2/2m$ ). In this configuration, the neural network only learns the potential energy $V(q)$ . When combined with relative coordinates, this formulation also enforces the conservation of angular momentum.

The authors define two primary model variants:

NHODEtot: Learns the total energy directly via a neural network.
NHODEpot: Learns only the potential energy, assuming a known kinetic energy form and particle masses.

Key Contributions

Novel Framework for Partial Observability: The paper introduces NHODE, the first framework to successfully combine Hamiltonian structure with Neural ODE training specifically to handle systems with completely unobserved state variables.
Demonstration of Physical Priors: The work systematically demonstrates that embedding physical structure (energy conservation, momentum conservation via symmetry) is not merely beneficial but essential for learning stable dynamics in partially observed, high-dimensional, and chaotic regimes.
Empirical Validation on Increasing Complexity: The method is evaluated on three systems of increasing complexity:
- A linear 1D two-mass, two-spring system.
- A nonlinear 2D triangular three-mass, three-spring system.
- The chaotic 3D three-body problem.
Initial Condition Learning: The framework is shown to effectively infer the initial conditions of hidden variables solely from observed trajectories, a task where purely data-driven baselines fail.

Results

The experiments compare NHODE against two baselines:

NODEvanilla: A standard Neural ODE.
NODEphys: A physics-informed Neural ODE that learns only momentum derivatives.

Findings:

Linear Systems: In the simple 1D mass-spring system, all models (including baselines) perform reasonably well, though NHODE models show slightly better stability.
Nonlinear Systems: In the 2D nonlinear mass-spring system, the performance gap widens. While NHODE models maintain low error, the baselines begin to diverge over longer time horizons.
Chaotic Systems: In the three-body problem, the distinction is stark. Purely data-driven baselines (NODEvanilla) and even the partially physics-informed NODEphys become unstable and diverge rapidly. Only the most physics-informed NHODE model (NHODEpot,rel), which enforces conservation of energy, linear momentum, and angular momentum, produces stable, non-diverging trajectories over long horizons.
Error Metrics: Across all examples, increasing the amount of embedded physical structure (e.g., moving from NHODEtot to NHODEpot, or from absolute to relative coordinates) systematically improves both prediction accuracy and long-horizon stability.

Significance and Claims

The paper claims that its primary contribution is demonstrating that physical knowledge embedded in a machine learning model allows for the accurate learning of both observed and latent parts of a partially observed system, a task where purely data-driven methods fail.

The authors emphasize that while the latent dynamics might not be uniquely identifiable without additional information (such as known system parameters), the NHODE framework learns a latent state that is physically consistent and yields accurate predictions for the observed variables. The work highlights that in chaotic regimes, where small perturbations lead to exponential divergence, the incorporation of strong physical priors (conservation laws) is critical to prevent the learned dynamics from "blowing up," thereby extending the useful prediction window.

The paper concludes that the NHODE framework is a generalizable tool for settings where dynamical systems are partially observed, physical structure is known, and stable temporal predictions are required, such as in industrial monitoring, robotics, and scientific experiments with limited sensor access.