High-fidelity physics simulations are powerful tools in the design and optimization of charged particle accelerators. However, the computational burden of these simulations often limits their use in practice for design optimization and experiment planning. It also precludes their use as on-line models tied directly to accelerator operation. We introduce an approach based on machine learning to create nonlinear, fast-executing surrogate models that are informed by a sparse sampling of the physics simulation. The models are O(10⁶)-O(10⁷) times more computationally efficient to execute. We also demonstrate that these models can be reliably used with multiobjective optimization to obtain orders-of-magnitude speedup in initial design studies and experiment planning. For example, we required 132 times fewer simulation evaluations to obtain an equivalent solution for our main test case, and initial studies suggest that between 330-550 times fewer simulation evaluations are needed when using an iterative retraining process. Our approach enables new ways for high-fidelity particle accelerator simulations to be used, at comparatively little computational cost.