In this paper, we consider a novel optimization design for multi-waveguide pinching-antenna systems, aiming to maximize the weighted sum rate (WSR) by jointly optimizing beamforming coefficients and antenna position. To handle the formulated non-convex problem, a gradient-based meta-learning joint optimization (GML-JO) algorithm is proposed. Specifically, the original problem is initially decomposed into two sub-problems of beamforming optimization and antenna position optimization through equivalent substitution. Then, the convex approximation methods are used to deal with the nonconvex constraints of sub-problems, and two sub-neural networks are constructed to calculate the sub-problems separately. Different from alternating optimization (AO), where two sub-problems are solved alternately and the solutions are influenced by the initial values, two sub-neural networks of proposed GML-JO with fixed channel coefficients are considered as local sub-tasks and the computation results are used to calculate the loss function of joint optimization. Finally, the parameters of sub-networks are updated using the average loss function over different sub-tasks and the solution that is robust to the initial value is obtained. Simulation results demonstrate that the proposed GML-JO algorithm achieves 5.6 bits/s/Hz WSR within 100 iterations, yielding a 32.7\% performance enhancement over conventional AO with substantially reduced computational complexity. Moreover, the proposed GML-JO algorithm is robust to different choices of initialization and yields better performance compared with the existing optimization methods.