Singular parameters and missing limits in neural PDE solvers
Neural tanh PDE solvers can miss finite limits as parameters diverge; kernel derivatives restore attainable approximations.
The paper analyzes neural PDE solvers that approach an accurate solution while their parameters diverge, so the limiting solution may have no finite representation and the best loss stays unattained. It connects missing limits in deep tanh networks to unbounded hidden parameters or increasingly redundant neurons. For translated-kernel models, the authors describe the missing functions and recover them by adding kernel derivatives, making the best approximation attainable under standard assumptions. Numerical studies follow the associated parameter growth and how completion affects PDE optimization.
- Neural PDE solvers can approach accuracy while parameters grow without bound.
- Missing tanh-network limits tie to unbounded or redundant neurons.
- Kernel derivatives complete translated-kernel models so the best approximation is attainable.
- Numerical studies track parameter growth and effects on PDE optimization.
From the original article
1 min read at arxiv.orgNeural solvers for partial differential equations (PDEs) can approach an accurate solution while their parameters grow without bound. In such cases, the limiting solution may have no finite representation in the chosen model, leaving the best loss unattained. Our analysis connects missing limits in deep neural tanh- networks to unbounded hidden parameters or increasingly redundant neurons.
By Daniel Fernández · arxiv.org
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