Neural Networks Learn Stable Physics-Based Models Straight From Data

Researchers at the University of Stuttgart have unveiled a machine-learning framework that builds mathematical models directly from simulation or measurement data while guaranteeing, by construction, two of the most prized properties in dynamical systems theory: passivity and stability. The method, presented in the open-access journal Computational Science and Engineering, addresses a long-standing tension in modeling. Traditional physics-based modeling demands considerable expert knowledge and development time, while purely data-driven approaches often produce black-box models that lack interpretability, structure, and sometimes reliability. The new framework, called ApHIN, short for Autoencoder-based port-Hamiltonian Identification Network, aims to deliver the best of both worlds.

The mathematical foundation of the approach is the port-Hamiltonian formulation, an energy-based way of describing physical systems. Port-Hamiltonian models represent system behavior through an energy function, a Hamiltonian, together with matrices that describe how energy is stored, exchanged, and dissipated. Because the formulation encodes an energy balance as a dissipation inequality, systems written in this form automatically satisfy passivity, meaning the energy stored in the system can never exceed the energy it started with plus the energy supplied through its ports. Under mild assumptions, this also yields Lyapunov stability, where solutions stay near equilibrium points, and boundedness of solutions. Port-Hamiltonian systems are additionally closed under interconnection, so coupling several of them produces another port-Hamiltonian system, a property that makes them attractive for modular multi-physics and network modeling.

The core innovation lies in how the Stuttgart team, led by Johannes Rettberg, Jonas Kneifl, Julius Herb, Patrick Buchfink, Jörg Fehr, and Bernard Haasdonk, forces a neural network to respect this structure. Rather than letting the network learn an arbitrary set of equations, the architecture constructs the port-Hamiltonian matrices through Cholesky-style factorizations of trainable weights. A skew-symmetric structure matrix is assembled from a strictly lower triangular weight matrix and its negative transpose, guaranteeing skew-symmetry. The dissipation and energy matrices are built as products of lower triangular factors with their transposes, which makes them symmetric and positive semidefinite or positive definite by construction. No matter what values the optimization settles on, the resulting model is a valid port-Hamiltonian system, and its passivity and stability guarantees come for free rather than being hoped for after the fact.

The framework comes in two variants. The first, pHIN, is intended for problems where the state variables are already low-dimensional. It identifies the port-Hamiltonian matrices directly from full-state observations and their time derivatives by minimizing a loss function that penalizes mismatches between observed and predicted dynamics. A parametric extension adds a deep neural network that maps parameter vectors, such as material properties, to the system matrices, allowing a single learned model to cover a whole family of systems. The second variant, ApHIN, wraps pHIN inside an autoencoder, a pair of neural networks in which an encoder compresses high-dimensional data into a small latent space and a decoder reconstructs the original states. In this latent space, the team identifies a linear port-Hamiltonian system, which is small and therefore rapidly computable, yet through the nonlinear coordinate transformation it can still describe systems whose behavior in the original coordinates is strongly nonlinear.

Training ApHIN involves a joint optimization of three objectives: a reconstruction loss ensuring the autoencoder faithfully compresses and reconstructs the data, the port-Hamiltonian loss ensuring the latent dynamics follow the structured equations, and a consistency loss comparing reference time derivatives in the original high-dimensional space with those implied by the decoded latent model. For very high-dimensional problems, the authors add a linear principal component analysis layer at the outermost encoder and decoder stages to cut computational cost, an option they call ApHIN linear when the nonlinear part becomes unnecessary altogether.

A theoretical contribution strengthens the case for the autoencoder design. The authors prove that, provided the autoencoder satisfies a projection property, meaning that encoding a decoded state returns the same latent state, the desirable system-theoretic properties transfer from the latent space into the original physical space. The decoded model defines a nonlinear port-Hamiltonian system on a submanifold of the state space, with the dissipation inequality, passivity, Lyapunov stability, and boundedness of solutions all preserved. Notably, a linear port-Hamiltonian system in the latent coordinates can correspond to a nonlinear one in the physical coordinates, because the decoder’s Jacobians make the reconstructed matrices state-dependent. The authors acknowledge that the projection property is generally not fulfilled exactly in practice, and they report numerically how closely it is approximated rather than enforcing it architecturally.

The numerical demonstrations span three examples of increasing difficulty. First, a mass-spring-damper chain with three masses served as a benchmark for pHIN alone. The physical parameters, masses and stiffnesses, were randomly sampled over wide ranges, and damping was set to zero to create a marginally stable reference system with purely imaginary poles. Compared with an unstructured linear identification network and with traditional matrix interpolation across parameters, the structured pHIN model preserved the expected sparsity and symmetry patterns, placed its eigenvalues correctly near the imaginary axis, and achieved a mean relative state error of roughly 0.023 on training data and 0.081 on test data, against 0.028 and 0.124 for the unstructured model and 0.135 and 0.539 for matrix interpolation. The unstructured approach even placed several poles in the right half of the complex plane, producing an unstable identified system whose dynamics could deviate qualitatively from the reference.

Second, a nonlinear mathematical pendulum with deflections up to sixty degrees demonstrated why the nonlinear coordinate transformation matters. Describing the pendulum as a linear port-Hamiltonian system in its Cartesian coordinates, or reducing those coordinates linearly with principal component analysis, failed to capture the dynamics. With the nonlinear autoencoder, however, ApHIN found a two-dimensional latent coordinate system in which the pendulum’s dynamics appear approximately linear, and the identified model tracked all states closely on test trajectories lasting twice as long as the training horizon, showing genuine extrapolation capability. The projection properties were satisfied to an accuracy on the order of ten to the minus four, and evaluating the full pipeline of encoding, integrating, and decoding took roughly 0.08 seconds.

The most demanding test was a high-dimensional disc brake model, a coupled thermo-mechanical system relevant to understanding brake squeal caused by frictional heat. The full-order finite element model, implemented in the commercial software Abaqus, has 15,750 degrees of freedom covering temperature, displacements, and velocities, and its operators are locked inside closed-source software, making intrusive reduction methods impossible. ApHIN nonetheless learned a low-dimensional port-Hamiltonian surrogate from simulation data spanning varying heat flux inputs and two material parameters, heat conductivity and density. On test simulations, the mean relative error stayed below two percent for temperature, below 1.5 percent for velocities, and below one percent for displacements, with the best trade-off at a latent dimension of twelve. An evaluation of the reduced model required about 0.34 seconds, enabling near real-time use. The largest local errors clustered around the heating area, but the overall heating and expansion behavior was captured well across varied parameters and inputs.

The authors are candid about the method’s limits. The framework is inherently restricted to data-generating systems that are themselves stable and passive, and it will fit poorly outside that class. The surrounding autoencoder is prone to errors when extrapolating beyond the training regime, and the approach relies on complete, high-fidelity data with accurate derivative information, so measurement noise could substantially degrade performance. Future work, the team writes, should extend the identification to incomplete and noisy observations, integrate constrained autoencoders that enforce the projection property, explore domain-specific reduction strategies for multi-physics systems, and generalize to nonlinear port-Hamiltonian systems in the latent space. The code and datasets have been released openly, inviting the modeling community to put the guarantees to the test.

Subject of Research: Data-driven identification of latent port-Hamiltonian systems using autoencoder-based neural networks

Article Title: Data-driven identification of latent port-Hamiltonian systems

Article References: Rettberg, J., Kneifl, J., Herb, J., Buchfink, P., Fehr, J., & Haasdonk, B. (2025). Data-driven identification of latent port-Hamiltonian systems. Computational Science and Engineering, 2(1), Article 4. https://doi.org/10.1007/s44207-025-00007-2

Image Credits: AI Generated

DOI: 10.1007/s44207-025-00007-2

Keywords: port-Hamiltonian systems, system identification, machine learning, autoencoders, model order reduction, passivity, stability, neural networks, multi-physics modeling, disc brake simulation, energy-based modeling, dimensionality reduction