Scientific computing applications heavily rely on multi-level loop nests operating on multidimensional arrays. This presents multiple optimization opportunities from exploiting parallelism to reducing data movement through prefetching and improved register usage. HPC frameworks often delegate fine-grained data movement optimization to compilers, but their low-level representations hamper analysis of common patterns, such as strided data accesses and loop-carried dependencies. In this paper, we introduce symbolic, inductive loop optimization (SILO), a novel technique that models data accesses and dependencies as functions of loop nest strides. This abstraction enables the automatic parallelization of sequentially-dependent loops, as well as data movement optimizations including software prefetching and pointer incrementation to reduce register spills. We demonstrate SILO on fundamental kernels from scientific applications with a focus on atmospheric models and numerical solvers, achieving up to 12$\times$ speedup over the state of the art.