Sparse autoencoder probes of Pythia models show concept activations jump at roughly 410M parameters and during mid-training, while early-layer features re-emerge at the output layer.
Scattering-Passive Structure-Preserving Finite Element Method for the Boundary Controlled Transport Equation with a Moving Mesh
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abstract
A structure-preserving Finite Element Method (FEM) for the transport equation in one- and two-dimensional domains is presented. This Distributed Parameter System (DPS) has non-collocated boundary control and observation, and reveals a scattering-energy preserving structure. We show that the discretized model preserves the aforementioned structure from the original infinite-dimensional system. Moreover, we analyse the case of moving meshes for the one-dimensional case. The moving mesh requires less states than the fixed one to produce solutions with a comparable accuracy, and it can also reduce the overshoot and oscillations of Gibbs phenomenon produced when using the FEM. Numerical simulations are provided for the case of a one-dimensional transport equation with fixed and moving meshes.
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Sparse autoencoder probes of Pythia models show concept activations jump at roughly 410M parameters and during mid-training, while early-layer features re-emerge at the output layer.