arXiv subjects

Bernhard Bachmann

Publications and source records attributed to Bernhard Bachmann.

7 recordsLinked to original sources

arXiv2026

Towards an FMI Layered Standard for DAE: Applications for Simulation and Optimization

The Functional Mock-up Interface (FMI) 3.0 standard for Model Exchange is restricted to hybrid ordinary differential equations, requiring any internal algebraic equations to be solved inside the Functional Mock-up Unit (FMU) before derivatives are returned to the importer. For models originating from, e.g. Modelica, this means that nonlinear algebraic equations must be solved through internal Newton iterations, which can reduce accuracy, increase computational cost, introduce hidden solver states, and cause robustness issues in downstream simulation and optimization workflows. In this article, we present a proposal for a layered standard, fmi-ls-dae, that exposes algebraic equations and their associated algebraic variables as part of a semi-explicit index-1 differential-algebraic equation. We describe the proposed extensions to the FMI XML schema and demonstrate the approach through prototype implementations: Dymola and CasADi generate FMUs that expose this semi-explicit index-1 formulation, while CasADi, FMIOPT, Simcenter Twin Activate, and MOO (the dynamic optimization tool of OpenModelica) import them for simulation and dynamic optimization. On an industrially relevant multilink suspension corner model, the proposed DAE-FMU formulation enables the optimization routine to converge on an optimal control problem on which the equivalent ODE-FMU fails to converge. We outline ongoing work towards supporting higher-index DAEs, consistent initialization, and event handling,

eess.SY↗

arXiv2025

Residual-Informed Learning of Solutions to Algebraic Loops

This paper presents a residual-informed machine learning approach for replacing algebraic loops in equation-based Modelica models with neural network surrogates. A feedforward neural network is trained using the residual (error) of the algebraic loop directly in its loss function, eliminating the need for a supervised dataset. This training strategy also resolves the issue of ambiguous solutions, allowing the surrogate to converge to a consistent solution rather than averaging multiple valid ones. Applied to the large-scale IEEE 14-Bus system, our method achieves a 60% reduction in simulation time compared to conventional simulations, while maintaining the same level of accuracy through error control mechanisms.

cs.LG↗

arXiv2025

Efficient Training of Physics-enhanced Neural ODEs via Direct Collocation and Nonlinear Programming

We propose a novel approach for training Physics-enhanced Neural ODEs (PeN-ODEs) by expressing the training process as a dynamic optimization problem. The full model, including neural components, is discretized using a high-order implicit Runge-Kutta method with flipped Legendre-Gauss-Radau points, resulting in a large-scale nonlinear program (NLP) efficiently solved by state-of-the-art NLP solvers such as Ipopt. This formulation enables simultaneous optimization of network parameters and state trajectories, addressing key limitations of ODE solver-based training in terms of stability, runtime, and accuracy. Extending on a recent direct collocation-based method for Neural ODEs, we generalize to PeN-ODEs, incorporate physical constraints, and present a custom, parallelized, open-source implementation. Benchmarks on a Quarter Vehicle Model and a Van-der-Pol oscillator demonstrate superior accuracy, speed, generalization with smaller networks compared to other training techniques. We also outline a planned integration into OpenModelica to enable accessible training of Neural DAEs.

cs.LG↗

arXiv2024

Multi-rate Runge-Kutta methods: stability analysis and applications

We present an approach for the efficient implementation of self-adjusting multi-rate Runge-Kutta methods and we introduce a novel stability analysis, that covers the multi-rate extensions of all standard Runge-Kutta methods and allows to assess the impact of different interpolation methods for the latent variables and of the use of an arbitrary number of sub-steps for the active variables. The stability analysis applies successfully to the model problem typically used in the literature for multi-rate methods. Furthermore,} we also propose a physically motivated model problem that can be used to assess stability to problems with purely imaginary eigenvalues and in situations closer to those arising in applications. Finally, we present an efficient implementation of multi-rate Runge-Kutta methods in the framework of the OpenModelica open-source modelling and simulation software. Results of several numerical experiments, performed with this implementation of the proposed methods, demonstrate the efficiency gains deriving from the use of the proposed multi-rate approach for physical modelling problems with multiple time scales.

math.NA↗

arXiv2021

On the choice of initial guesses for the Newton-Raphson algorithm

The initialization of equation-based differential-algebraic system models, and more in general the solution of many engineering and scientific problems, require the solution of systems of nonlinear equations. Newton-Raphson's method is widely used for this purpose; it is very efficient in the computation of the solution if the initial guess is close enough to it, but it can fail otherwise. In this paper, several criteria are introduced to analyze the influence of the initial guess on the evolution of Newton-Raphson's algorithm and to identify which initial guesses need to be improved in case of convergence failure. In particular, indicators based on first and second derivatives of the residual function are introduced, whose values allow to assess how much the initial guess of each variable can be responsible for the convergence failure. The use of such criteria, which are based on rigorously proven results, is successfully demonstrated in three exemplary test cases.

math.NA↗

arXiv2015

Efficient algorithm for computing large scale systems of differential algebraic equations

In many mathematical models of physical phenomenons and engineering fields, such as electrical circuits or mechanical multibody systems, which generate the differential algebraic equations (DAEs) systems naturally. In general, the feature of DAEs is a sparse large scale system of fully nonlinear and high index. To make use of its sparsity, this paper provides a simple and efficient algorithm for computing the large scale DAEs system. We exploit the shortest augmenting path algorithm for finding maximum value transversal (MVT) as well as block triangular forms (BTF). We also present the extended signature matrix method with the block fixed point iteration and its complexity results. Furthermore, a range of nontrivial problems are demonstrated by our algorithm.

math.NA↗

arXiv2015

Index reduction of differential algebraic equations by differential algebraic elimination

High index differential algebraic equations (DAEs) are ordinary differential equations (ODEs) with constraints and arise frequently from many mathematical models of physical phenomenons and engineering fields. In this paper, we generalize the idea of differential elimination with Dixon resultant to polynomially nonlinear DAEs. We propose a new algorithm for index reduction of DAEs and establish the notion of differential algebraic elimination, which can provide the differential algebraic resultant of the enlarged system of original equations. To make use of structure of DAEs, variable pencil technique is given to determine the termination of differentiation. Moreover, we also provide a heuristics method for removing the extraneous factors from differential algebraic resultant. The experimentation shows that the proposed algorithm outperforms existing ones for many examples taken from the literature.

cs.SC↗