text: 205,51,1262793302388113411,2933022295,Alex Beatson,"New paper: ""Learning Composable Energy Surrogates for PDE Order Reduction"", with Jordan Ash, @geoffrey_roeder, @TianjuXue, @ryan_p_adams: <LINK>. We use NNs to amortize solving hard mechanical meta-material PDEs, training only on data from small subdomains. 1/8 <LINK> How? Solutions to hyperelasticity PDEs (and many others!) are minimizers of a total energy, which is an integral over the domain. We train NNs to predict the minimal energy in a component from a reduced-basis solution (here, a displacement field) on the component boundary. 2/8 When components are composed into a larger domain, the total energy is a sum of component energies. We solve the PDE in a reduced basis of component boundaries, minimizing the sum of NN surrogate energies. We only generate supervised data (via FEA) on the small components. 3/8 <LINK> When solving PDEs with FEA, fine geometric features (such as metamaterial pores) require a fine mesh. Part of the trick here is feeding parametric representations of geometry to our component-level energy surrogate, avoiding representing geometry with a fine mesh. 4/8 To get accurate surrogates, we embed known structure (invariance to rigid-body transforms and a linear-elastic structural prior) into the NN surrogate, and perform Sobolev training on energy derivatives and Hessian-vector products as well as energy values. 5/8 The learned surrogate can solve composed systems orders-of-magnitude faster than a finite element model of similar accuracy, run on the same CPU. 6/8 <LINK> Limitation: we use data augmentation to capture displacements the components encounter in practice, so you have to pick a distribution of macroscopic problems to amortize. Perhaps inevitable, but we’re looking into relaxing this via active learning and learning robust models. 7/8 Mechanical meta-materials like the lattices we study hold great promise for engineering design (see <LINK>), but are hard to simulate due to geometric complexity and nonlinear behavior. We think there’s a lot of room for ML to help harness their potential. 8/8",https://arxiv.org/abs/2005.06549,"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 with individual components. Using dataset aggregation to choose training boundary conditions allows us to learn energy surrogates which produce accurate macroscopic behavior when composed, accelerating simulation of parametric meta-materials. ",Learning Composable Energy Surrogates for PDE Order Reduction,8,"['New paper: ""Learning Composable Energy Surrogates for PDE Order Reduction"", with Jordan Ash, @geoffrey_roeder, @TianjuXue, @ryan_p_adams: <LINK>. We use NNs to amortize solving hard mechanical meta-material PDEs, training only on data from small subdomains. 1/8 <LINK>', 'How? Solutions to hyperelasticity PDEs (and many others!) are minimizers of a total energy, which is an integral over the domain. We train NNs to predict the minimal energy in a component from a reduced-basis solution (here, a displacement field) on the component boundary. 2/8', 'When components are composed into a larger domain, the total energy is a sum of component energies. We solve the PDE in a reduced basis of component boundaries, minimizing the sum of NN surrogate energies. We only generate supervised data (via FEA) on the small components. 3/8 https://t.co/0NrQfQitq9', 'When solving PDEs with FEA, fine geometric features (such as metamaterial pores) require a fine mesh. Part of the trick here is feeding parametric representations of geometry to our component-level energy surrogate, avoiding representing geometry with a fine mesh. 4/8', 'To get accurate surrogates, we embed known structure (invariance to rigid-body transforms and a linear-elastic structural prior) into the NN surrogate, and perform Sobolev training on energy derivatives and Hessian-vector products as well as energy values. 5/8', 'The learned surrogate can solve composed systems orders-of-magnitude faster than a finite element model of similar accuracy, run on the same CPU. 6/8 https://t.co/Qmwu01ao9M', 'Limitation: we use data augmentation to capture displacements the components encounter in practice, so you have to pick a distribution of macroscopic problems to amortize. Perhaps inevitable, but we’re looking into relaxing this via active learning and learning robust models. 7/8', 'Mechanical meta-materials like the lattices we study hold great promise for engineering design (see https://t.co/GmrQEiwwuA), but are hard to simulate due to geometric complexity and nonlinear behavior. We think there’s a lot of room for ML to help harness their potential.  8/8']",20,05,2060