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Structure-Preserving Data-Driven Identification of Port-Hamiltonian Differential-Algebraic Systems
Tom Zwerschke, Matthias Ehrhardt, Michael Günther, Claudia Totzeck
TL;DR
Identifying pH-DAEs from input-output data is less developed than pH-ODE identification, while retaining pH structure matters for preserving important system properties. This paper develops an index-1 structure-preserving identification approach, and experiments report accurate reference-system surrogates that generalize to independent inputs.
Problem
When detailed subsystem models are unavailable, identifying pH-DAEs from input-output data is less developed, yet retaining pH structure matters because important properties may otherwise be lost.
Source: §1 · pp. 2–3Method
The approach parameterizes linear index-1 pH-DAEs, preserves their structure and algebraic constraints in discretization, and uses an adjoint gradient for reduced parameter optimization.
Source: §1 · p. 3, §7 · p. 26Results
Experiments show accurate input-output reproduction of reference systems, with cross-validation indicating generalization to independent inputs and potential for lower-dimensional surrogates.
Source: §1 · p. 3, §7 · p. 26Takeaways & Limitations
The experiments support the approach as a source of structure-preserving pH-DAE surrogate models, including lower-dimensional models and predictions for unseen inputs.
Source: §1 · p. 3, §7 · p. 26The study focuses on linear index-1 systems; extending the approach to higher-index and nonlinear pH-DAEs, noisy or incomplete measurements, and uncertainty quantification remains future work.
Source: §7 · p. 26
Abstract
from arXiv · showhide
We present a data-driven approach to identifying linear index-1 differential-algebraic pH systems (pH-DAEs) based on input-output measurements. In comparison to the identification of port-Hamiltonian (pH) systems, the algebraic constraint and the index condition pose additional challenges. First, we establish a structure-preserving formulation of the considered pH-DAE class and derive an implicit midpoint discretization that preserves the algebraic constraints and discrete dissipation inequality. We formulate the identification problem as a regularized least-squares minimization problem subject to the pH-DAE dynamics. Exploiting the index-1 structure, we reduce the constrained problem to an unconstrained optimization problem over the system parameters while preserving the port-Hamiltonian structure. Next, we derive an adjoint-based formulation to efficiently evaluate the gradient of the resulting reduced cost functional. This enables us to use gradient-based optimization methods for parameter estimation. Under suitable assumptions on the admissible parameter set, the existence of a minimizer is established. Numerical experiments demonstrate that the proposed approach can identify surrogate pH-DAE systems that accurately reproduce the input-output behavior of reference systems. Further investigations show the approach's potential for identifying reduced-order surrogate models. Cross-validation with independent input signals confirms the predictive capability of the identified models.
1. Introduction
The introduction motivates structure-preserving identification for systems known only through input-output data and extends pH-ODE calibration to linear index-1 pH-DAEs. The approach addresses algebraic constraints and the index condition, and experiments report accurate surrogate behavior, including on independent inputs.
1. Introduction
- Because detailed subsystem models may be unavailable, data-driven identification can construct surrogates from measurements while retaining pH structure.Without that structure, important physical-system properties may be lost.Source: §1 · p. 2
- Prior pH-ODE calibration embeds pH constraints in time-domain optimization, while pH-DAE identification is less developed and one existing approach requires state information.That approach uses Gaussian processes and a nonparametric strategy.Source: §1 · pp. 2–3
- The paper extends adjoint-based calibration to linear index-1 pH-DAEs, using structure-preserving discretization and index-1 reduction to optimize system parameters.An adjoint gradient supports gradient-based optimization, and a minimizer exists under suitable assumptions.Source: §1 · p. 3
- Identified pH-DAE surrogates reproduce reference input-output behavior, with cross-validation indicating useful predictions for independent inputs.The experiments also investigate lower-dimensional structure-preserving surrogates.Source: §1 · p. 3
2. Port-Hamiltonian Systems
The section presents port-Hamiltonian systems as models organized around energy, dissipation, and external interaction. It also describes structure-preserving interconnection and a discrete-gradient route to an implicit midpoint scheme for quadratic Hamiltonians.
2. Port-Hamiltonian Systems
- A port-Hamiltonian system uses a skew-symmetric matrix J, a positive semidefinite dissipation matrix R, and a port matrix B for input-output coupling.The input and output signals are denoted u and y.Source: §2 · p. 4
- For a linear system, the quadratic Hamiltonian has a symmetric positive semidefinite matrix Q, consistent with its interpretation as nonnegative energy.The Hamiltonian equations for mechanical systems use generalized coordinates and momenta.Source: §2 · pp. 3–4
- Port-Hamiltonian systems satisfy a dissipation inequality, and suitable interconnections again yield a monolithic pH system.A discrete-gradient method satisfies the discrete dissipation inequality; for a quadratic Hamiltonian, its midpoint version coincides with the implicit midpoint rule.Source: §2 · pp. 4–5
3. Port-Hamiltonian Differential-Algebraic Systems
This section defines the considered linear pH-DAE class and focuses on a semi-explicit index-1 form. It explains how the algebraic variables are determined and how implicit midpoint discretization preserves constraints and discrete dissipation under stated conditions.
3. Port-Hamiltonian Differential-Algebraic Systems
- The considered linear pH-DAE class is defined through structural conditions on its differential-algebraic operator, including positive semidefiniteness.Its underlying quadratic Hamiltonian is introduced as part of the system description.Source: §3 · pp. 5–6
- The system is put in semi-explicit form with differential variables x1 and algebraic variables x2; index 1 requires J22 − R22 to be regular so x2 can be solved.When Q is regular, a linear pH-DAE can be transformed into this form.Source: §3 · p. 6
- Eliminating the algebraic part yields a closed-form differential equation for x1, while consistency determines the initial value of x2 from x1,0.The initial value of the differential variables can be chosen arbitrarily.Source: §3 · p. 6
- For the transformed index-1 system, the implicit midpoint rule is well-defined and preserves algebraic constraints and discrete dissipation under certain conditions.The discretization evaluates the state at the midpoint and uses the midpoint input.Source: §3 · pp. 6–7
4. Identification Problem
The section formulates identification of linear index-1 pH-DAEs from input-output data as a parameter optimization problem constrained by the system dynamics. It exploits the index-1 structure to define an admissible parameter set and establish conditions for a global minimizer.
4. Identification Problem
- The data are modeled as input-output measurements of an unknown linear index-1 pH-DAE, with discrete signals interpolated when needed.The internal-state dimension n is assumed specified; different numbers r of differential variables can be assessed by comparing costs and testing on a second input.Source: §4 · pp. 7–8
- The admissible parameters enforce skew-symmetric J, positive-semidefinite R, and a uniformly nonsingular algebraic block J22−R22.The smallest singular value of J22−R22 is bounded below by a positive threshold.Source: §4 · pp. 7–8
- The calibration minimizes a cost over system matrices and initial data subject to the state equation, and the state constraint is handled implicitly in the reduced formulation.The state operator is weakly continuous, a property used in the existence argument.Source: §4 · pp. 8, 10, 11
- For each candidate parameter vector, the state equation has a unique solution that depends continuously on the data.The formulation uses the resulting solution operator to express the identification objective as a reduced cost.Source: §4 · pp. 9–10
- A global minimizer exists when λ > 0 or the admissible parameter set Uad is bounded.The minimizer need not be unique because the reformulated problem is generally non-convex.Source: §4 · p. 11
5. Numerical Optimization
The section develops numerical strategies for optimizing the pH-DAE parameters while maintaining the index-1 regularity condition. It then derives an adjoint-based gradient for the reduced objective and describes an iterative descent procedure.
5. Numerical Optimization
- The parameterization J=K−K⊤ and R=LL⊤ preserves skew-symmetry and positive semidefiniteness while turning the search into an unconstrained parameter optimization.It introduces more degrees of freedom and loses the linear structure; the regularity condition still couples J and R.Source: §5 · p. 12
- A log-determinant barrier penalizes near-singular J22−R22, but requires a sequence μ_i→0 and repeated full inversions during gradient evaluation.As the determinant approaches zero, the penalty tends to infinity; gradient evaluation requires solving a matrix equation to obtain the inverse.Source: §5 · pp. 12–13
- The heuristic rejects a proposed step when the state equation cannot be solved, preventing iterates from reaching singular algebraic blocks.The shared-kernel condition ker(J22)∩ker(R22) rarely occurred in the numerical examples.Source: §5 · p. 13
5.1. The Gradient Descent Method
- The adjoint method computes the reduced-cost gradient by solving the state equation forward and the adjoint equation backward, then using both solutions to choose a descent direction.The adjoint is itself an index-1 pH-DAE and is solved with terminal condition p1(T)=0.Source: §5.1 · pp. 13, 14, 16, 17
6. Numerical Results
The experiments vary the number of differential variables in synthetic and RCL-ladder systems, then assess fitted outputs using both calibration data and independent test signals. They report accurate fits for many model orders, while showing that order choice affects results and that the RCL ladder requires a DAE representation.
6.1. Synthetic data
- For rref=4, fitted outputs matched well except at r=1; r=3 had the lowest calibration cost and maximum-norm error among tested orders.The study tested r=1 through r=5.Source: §6.1 · p. 19, Table 1 · p. 20, Figure 5 · p. 20
- In the cross-validation test with rref = 4, the r = 5 fitted system had the largest error, while r = 4 had the smallest error and was closest to the actual value; r = 2 and 3 also matched well.Source: §6.1 · p. 20, Table 2 · p. 21, Figure 6 · p. 21
- >99% cost reduction occurred for every tested order except r=1 when rref=2; fitted outputs closely matched, and r=2 performed best in cross-validation.The cross-validation result identifies r=2 as having the lowest cost and smallest errors.Source: §6.1 · p. 21, Table 4 · p. 23, Figure 8 · p. 23
- The authors report that cross-validation depends heavily on r and that different initial data can yield cost-minimizing surrogates that respond differently to the test signal.The identification problem does not have a unique solution.Source: §6.1 · p. 21
6.2. Model order reduction test with a RCL Ladder Network
- Across 12 test cases with N ∈ {100, 50, 20} and r/n ∈ {1, 0.9, 0.6, 0.3}, all costs were below 1e−5; relative errors were below 1% except in three cases with N=r, and outputs matched well only when r<n.Source: §6.2 · p. 24
- In RCL-ladder cross-validation, the ODE system could not represent the reference, while the other tested systems produced close outputs.The cross-validation outputs are compared with the original system.Source: §6.2 · p. 24, Figure 13 · p. 25, Table 6 · p. 26
7. Conclusion
The paper presents an input-output identification approach for linear index-1 pH-DAEs that retains pH structure and uses index-1 reduction with an adjoint gradient. Experiments show accurate reference-system reproduction, predictive capability on independent signals, and potential for lower-dimensional surrogates; extensions to broader settings remain future work.
7. Conclusion
- The index-1 structure reduces identification to unconstrained optimization, and an adjoint formulation efficiently supplies the reduced cost functional's gradient.The parameterization incorporates pH structure so identified models retain relevant structural properties.Source: §7 · p. 26
- Experiments show accurate input-output reproduction of reference systems, generalization to independent input signals, and potential for lower-dimensional structure-preserving surrogates.The reported cross-validation uses independent signals.Source: §7 · p. 26
- Future work targets higher-index and nonlinear pH-DAEs, noisy or incomplete measurements, uncertainty quantification, and systematic identification with structure-preserving model reduction.The stated scope also includes large-scale multiphysics applications.Source: §7 · p. 26