In this paper we present a lattice structure optimization approach by leveraging the capabilities of neural networks for implicit geometry representation. We employ the Deep Signed Distance Field (DeepSDF) method, where a continuous and low-dimensional latent space is introduced to encode the geometric information. In contrast to traditional topology optimization methods, this allows the restriction of the design space to specific geometries. In our case, the latent space is used to represent the geometry of different unit cells, that are stacked to form a lattice structure. Moreover, continuously varying the latent vector over the structure allows a functional grading and optimization. Unlike other lattice-structure optimization methods, we neither assume a large separation of scale nor periodicity. Instead, we perform a full-scale finite element analysis at each optimization step. The required mesh is obtained by a differentiable extension of the dual marching cubes algorithm, which enables gradient-based optimization.