Singular parameters and missing limits in neural PDE solvers
Some neural PDE solvers can converge toward solutions their own finite parameterization cannot actually attain.
The paper analyzes cases where accurate PDE approximations require hidden parameters to grow without bound or neurons to become redundant. It ties those “missing limits” to deep tanh networks and to translated-kernel models. For the kernel class, it says the missing functions can be recovered by adding kernel derivatives, making the best approximation attainable under standard assumptions. Numerical studies track the parameter growth and test how that completion changes PDE optimization. ArXiv · AI/CL/LG's note
The paper analyzes cases where accurate PDE approximations require hidden parameters to grow without bound or neurons to become redundant. It ties those “missing limits” to deep tanh networks and to translated-kernel models. For the kernel class, it says the missing functions can be recovered by adding kernel derivatives, making the best approximation attainable under standard assumptions. Numerical studies track the parameter growth and test how that completion changes PDE optimization. ArXiv · AI/CL/LG's note
score 4