다중해를 갖는 주파수 응답곡선의 물리정보 신경망 해석: 초기값 정식화의 한계와 호길이 연속법

키워드

확장요약문

Where the frequency response curve of a nonlinear vibration folds back through saddle-node bifurcations, three steady solutions coexist at one excitation frequency, one of them an unstable saddle. A physics-informed neural network (PINN) posed as an initial-value problem finds only the solution its initial condition selects, never the unstable one. We examine this limitation on the harmonically excited rectangular plate with 1∶1 internal resonance (Chang, Bajaj and Krousgrill, Nonlinear Dyn. 4, 1993), under a forcing for which the response is multi-valued but the internal resonance is not yet active.

The three formulations use the same network (four tanh hidden layers of width 64) but differ in the equations they solve, that is, in the physical constraint (Table 1). Formulation I solves the original non-autonomous equations of motion, where fast oscillation and slow amplitude variation coexist, as an initial-value problem with Fourier-feature input against spectral bias. Formulation II solves the averaged autonomous slow floẇ 𝑧= 𝑓(𝑧; 𝜔) the same way, without the fast carrier. Formulation III removes time: its unknown is the response curve itself, the set of (𝑢1, 𝑣1, 𝜔) satisfying the single-mode fixed-point condition 𝑓= 0. The network maps arclength 𝑠to the unit tangent, and position follows by integration.

Away from the saddle-node bifurcations all three reproduce the stable solutions within 0.4% (Fig. 1). Near a bifurcation, however, the initial-value formulations are limited by the dynamics, not by network capacity: the distance between the stable solution and the saddle shrinks and the slowest decay rate decreases, and Formulation I misses the two points nearest the bifurcation. Formulation II, free of the fast carrier, finds every stable solution, but as an initial-value problem it cannot reach the unstable ones, and its accuracy is bounded by the averaging error. Formulation III obtains the entire curve in one training run, including both bifurcations and the unstable segment that no initial-value method can reach. Classical continuation is much faster for this forward problem; the advantage of the PINN lies in inverse problems with unknown parameters.