We present Tensor-GaLore, a novel method for efficient training of neural networks with higher-order tensor weights. Many models, particularly those used in scientific computing and computer vision, employ tensor-parameterized layers to capture complex, high-dimensional relationships. However, these tensor structures lead to significant memory requirements during training. Our method addresses this memory challenge through low-rank subspace optimization using Tucker decomposition, overcoming limitations of previous approaches restricted to matrix-parameterized weights, including those operating on complex-valued data. We showcase its effectiveness on Fourier Neural Operators (FNOs), a class of models crucial for solving partial differential equations. Across various PDE tasks, we achieved performance gains ranging from 11\% to 50\% better generalization while reducing optimizer memory usage by up to 76\%. These consistent improvements, coupled with substantial memory savings across AI for science, demonstrate Tensor-GaLore's potential.