72 M. J. Szydlowski et al. 9.2 Background LNNs [16] provide a way to parametrize an arbitrary Lagrangian of a dynamical system. The well-established Lagrangian mechanics are based on the principle of least action [17] given by (9.1) .S= t1 t0 (T (qt , ˙qt ) −V (qt ))dt = t1 t0L (qt , ˙qt )dt (9.1) where. Sis the scalar value of the action, .q, ˙q are generalized coordinates of the system, and . Lis the Lagrangian defined as the difference between kinetic energy T and potential energy V. The Euler-Lagrange equation . d dt ∂L ∂˙q = ∂L ∂q (9.2) can be expanded to the following equation of motion (EOM) in vector form . ¨q = ∇˙q∇ T q L −1 ∇qL− ∇q∇ T ˙q L ˙q , (9.3) as shown by Cranmer et al. [16]. The resulting EOM enables calculating . ¨qt for given a set of coordinates .qt , ˙qt . In this work, we consider the case where . L is an arbitrary Lagrangian function parametrized by a neural network. As integrating (9.3) yields the dynamics of the system, it is, therefore, possible to use it in a loss function that minimizes the discrepancy between model predictions, . ¨qt , and experimental measurements, . ¨qt true. It is also worth noting that modern automatic differentiation frameworks have no problem with deriving the Jacobian, and hessian of . Lfound in Eq. (9.3). 9.3 Introducing Nonconservative Forces To expand the use of LNN to an engineering setting, let us consider a nonconservative system, with the presence of damping forces D, and other external excitation forces F. For such a system, one can modify (9.2), and write the following expression: . d dt ∂L ∂˙q − ∂L ∂q + ∂D ∂˙q + ∂F ∂q = 0. (9.4) Similarly, to how Eq. (9.3) was achieved, expanding (9.4) yields the following expression: . ¨q = ∇˙q∇ T q L −1 ∇qL+∇qE− ∇q∇ T ˙q L+∇˙qF ˙q . (9.5) Here both . Land the dissipative forces Dare parametrized by a neural network, and the external forces ∇q Fare assumed to be measured directly from the experiment. 9.4 Experimental Analysis To test the application of the extended LNN, a set of numerical experiments was conducted. A damped nonlinear mechanical system excited externally by a harmonic force F was simulated. The system comprised of a mass m, a linear k, a nonlinear cubic spring k3, and a proportional damper c. The training data for the proposed presented method was simulated numerically using the following parameters m= 0.64, k =1, k3 =0.5, and c = 0.0064, and for a set of different excitation frequencies as well as forces. Solutions were obtained using shooting with pseudo-arc length continuation methods [3] by fixing one parameter, the excitation frequency, or forcing amplitude and using the other one as a continuation parameter. This provided
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