cs.CESep 24, 2026

AFT Neural Function Approximators for 1D Nonlinear Force Laws

Authors: Miriam Goldack, Johann Groß, Malte Krack, Merten Stender

Organizations: Technische Universität Berlin, Chair of Cyber-Physical Systems in Mechanical Engineering, Straße des 17. Juni 135, 10623 Berlin, Germany · University of Stuttgart, Institute of Aircraft Propulsion Systems, Pfaffenwaldring 6, 70569 Stuttgart, Germany

Read PDF
Ask AI
Cite

Copy citation

DOINot available for this paper

Abstract

Nonlinear contacts and friction strongly influence the vibration response of assembled structures, but their accurate numerical treatment is computationally demanding. The harmonic balance method is widely used to compute periodic steady-state responses, yet the required alternating frequency-time scheme becomes costly for nonsmooth and hysteretic nonlinearities and must be repeated throughout the nonlinear solution process. Here we show that this procedure can be replaced by neural networks that directly map displacement Fourier coefficients to nonlinear force coefficients and provide the corresponding Jacobian through automatic differentiation. The surrounding solver and continuation algorithms remain unchanged for the computation of frequency response curves. The neural networks exclusively learn individual nonlinear elements rather than complete system responses. Physics-based nondimensionalization and phase normalization facilitate the learning process and enable a single trained network to cover a wide range of parameter combinations. Building on the cubic spring, unilateral spring, and Jenkins elements considered here, the approach points toward a reusable library of nonlinear-element surrogates that can be combined in arbitrary number and location within a mechanical system. By bypassing the iterative force evaluation in time domain, the method offers favorable computational scaling for high-resolution analyses and systems with many nonlinear elements.

Figures & tables

Figure 1: Comparison of the AFT- and SFN-based workflows within the HBM. Green and red indicate vectorizable and non-vectorizable operations, respectively; solid and dashed borders denote the SFN and AFT workflows.
Cubic springUnilateral springJenkins element
   
   
Table 1: Schematics of a single-mass damped oscillator with three different nonlinearities, force-displacement diagrams, and nonlinear force history for different numbers of time samples NN .
 Cubic springUnilateral springJenkins element
Force coefficients: global relative L2L^{2} -error εf^,2glob\varepsilon_{\hat{\mathbf{f}},2}^{\mathrm{glob}}
 0.001 350.001\,350.136 350.136\,350.003 140.003\,14
Force coefficients: component-wise normalized RMSE
 A1A_{1} 0.001 340.001\,34 B1B_{1} 0.882 920.882\,92 A3A_{3} 0.004 310.004\,31 B3B_{3} 1.684 961.684\,96A0A_{0} 0.254 980.254\,98 A1A_{1} 0.234 560.234\,56 B1B_{1} 0.654 520.654\,52 A2A_{2} 0.274 310.274\,31 B2B_{2} 0.300 110.300\,11A1A_{1} 0.002 920.002\,92 B1B_{1} 0.004 580.004\,58 A3A_{3} 0.031 470.031\,47 B3B_{3} 0.015 290.015\,29
Jacobian: mean pointwise relative Frobenius norm error εˉJ,F\bar{\varepsilon}_{\mathbf{J},F}
 0.000 580.000\,580.160 130.160\,130.042 570.042\,57
Table 2: Comparison of the force-coefficient and Jacobian predictions along the considered frequency-response continuation. All global error metrics are evaluated using the same nΩn_{\Omega} points on the solution curve and identical inputs for the AFT and SFN evaluations.
Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Test caseSamplesCoefficientSampling rule
Cubic spring10510^{5}a1′a_{1}^{\prime}U[0, 5]\mathcal{U}[0,\,5]
a3′a_{3}^{\prime}U[−5, 5]\mathcal{U}[-5,\,5]
b3′b_{3}^{\prime}U[−5, 5]\mathcal{U}[-5,\,5]
Unilateral spring10610^{6}a0∗,′a_{0}^{*,\prime}U[−2, 2]\mathcal{U}[-2,\,2]
a1∗,′a_{1}^{*,\prime}U[0, 2]\mathcal{U}[0,\,2]
a2∗,′a_{2}^{*,\prime}U[−2, 2]\mathcal{U}[-2,\,2]
Appendix
Table 3: Sampling domains and dataset sizes used for the generation of the SFN training data.
RegimeSamplesCoeff.Sampling rule
Open/ onset2500a0∗,′a_{0}^{*,\prime}0
a1∗,′a_{1}^{*,\prime}N[0.01, 1.1](1.0, 0.3)\mathcal{N}_{[0.01,\,1.1]}(1.0,\,0.3)
a2∗,′a_{2}^{*,\prime}0
b2∗,′b_{2}^{*,\prime}0
Contact- active7500a0∗,′a_{0}^{*,\prime}N[−0.2, 0.2](−0.05, 0.04)−0.6(a1∗,′−1)\begin{aligned} \mathcal{N}_{[-0.2,\,0.2]}(-0.05,\,0.04)\\ -0.6(a_{1}^{*,\prime}-1)\end{aligned}
a1∗,′a_{1}^{*,\prime}N[1.0, 2.0](1.5, 0.3)\mathcal{N}_{[1.0,\,2.0]}(1.5,\,0.3)
Appendix
Table 4: Application-specific sampling strategy for unilateral spring training data.
 Cubic springUnilateral springJenkins element
Network typeFeedforward neural networkFeedforward neural networkFeedforward neural network
Input dimension343
Output dimension454
Hidden layers355
Neurons per hidden layer128128128
ActivationGELUGELUGELU
Appendix
Table 5: Architecture and training strategy of the SFNs used for the three application cases.
Test caseParametersMethodncontn_{\mathrm{cont}}∑nit\sum n_{\mathrm{it}}n‾it\overline{n}_{\mathrm{it}}∑nFC\sum n_{\mathrm{FC}}
Cubic springk3=0.1,  F0=0.1k_{3}=0.1,\;F_{0}=0.1AFT28541.9382
SFN28541.9382
k3=0.1,  F0=0.2k_{3}=0.1,\;F_{0}=0.2AFT44851.93129
SFN44851.93129
k3=0.1,  F0=0.3k_{3}=0.1,\;F_{0}=0.3AFT571061.86163
SFN571061.86163
Appendix
Table 6: Solver statistics for the adaptive continuation runs. The AFT-based formulation serves as reference.

Explore similar work

CardsList

  1. Equilibria for Networks of Linear Translational Springs

    Sep 2, 2026Luke Oeding, Ethan Clayton, Jackson Elsea +2SplinesEquilibrium

    Auburn University, Department of Mathematics and Statistics, Auburn, AL, USA · New York University, Courant Institute of Mathematical Sciences, New York, NY, USA · Air Force Research Laboratory, Eglin Air Force Base, Florida, USA

  2. One-shot learning for the complex dynamical behaviors of weakly nonlinear forced oscillators

    Apr 16, 2026Teng Ma, Luca Rosafalco, Wei Cui +2System IdentificationNonlinear Dynamics

  3. Data-driven Koopman mode approximation: A neural power iteration algorithm

    Aug 27, 2026Guillaume O. Berger, Raphaël M. JungersKoopman OperatorNonlinear Dynamics

    ICTEAM, UCLouvain, Louvain-la-Neuve, Belgium