In design, fabrication, and control problems, we are often faced with the task of synthesis, in which we must generate an object or configuration that satisfies a set of constraints while maximizing one or more objective functions. The synthesis problem is typically characterized by a physical process in which many different realizations may achieve the goal. This many-to-one map presents challenges to the supervised learning of feed-forward synthesis, as the set of viable designs may have a complex structure. In addition, the non-differentiable nature of many physical simulations prevents efficient direct optimization. We address both of these problems with a two-stage neural network architecture that we may consider to be an autoencoder. We first learn the decoder: a differentiable surrogate that approximates the many-to-one physical realization process. We then learn the encoder, which maps from goal to design, while using the fixed decoder to evaluate the quality of the realization. We evaluate the approach on two case studies: extruder path planning in additive manufacturing and constrained soft robot inverse kinematics. We compare our approach to direct optimization of the design using the learned surrogate, and to supervised learning of the synthesis problem. We find that our approach produces higher quality solutions than supervised learning, while being competitive in quality with direct optimization, at a greatly reduced computational cost.

Optimizing the parameters of partial differential equations (PDEs), i.e., PDE-constrained optimization (PDE-CO), allows us to model natural systems from observations or perform rational design of structures with complicated mechanical, thermal, or electromagnetic properties. However, PDE-CO is often computationally prohibitive due to the need to solve the PDE—typically via finite element analysis (FEA)—at each step of the optimization procedure. In this paper we propose amortized finite element analysis (AmorFEA), in which a neural network learns to produce accurate PDE solutions, while preserving many of the advantages of traditional finite element methods. This network is trained to directly minimize the potential energy from which the PDE and finite element method are derived, avoiding the need to generate costly supervised training data by solving PDEs with traditional FEA. As FEA is a variational procedure, AmorFEA is a direct analogue to popular amortized inference approaches in latent variable models, with the finite element basis acting as the variational family. AmorFEA can perform PDE-CO without the need to repeatedly solve the associated PDE, accelerating optimization when compared to a traditional workflow using FEA and the adjoint method.

Meta-materials are an important emerging class of engineered materials in which complex macroscopic behaviour--whether electromagnetic, thermal, or mechanical--arises from modular substructure. Simulation and optimization of these materials are computationally challenging, as rich substructures necessitate high-fidelity finite element meshes to solve the governing PDEs. To address this, we leverage parametric modular structure to learn component-level surrogates, enabling cheaper high-fidelity simulation. We use a neural network to model the stored potential energy in a component given boundary conditions. This yields a structured prediction task: macroscopic behavior is determined by the minimizer of the system's total potential energy, which can be approximated by composing these surrogate models. Composable energy surrogates thus permit simulation in the reduced basis of component boundaries. Costly ground-truth simulation of the full structure is avoided, as training data are generated by performing finite element analysis of individual components. Using dataset aggregation to choose training data allows us to learn energy surrogates which produce accurate macroscopic behavior when composed, accelerating simulation of parametric meta-materials.