Automating Variational Differentiation
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910499559440384 |
|---|---|
| author | Li, Kangbo Damle, Anil |
| author_facet | Li, Kangbo Damle, Anil |
| contents | Many problems in Physics and Chemistry are formulated as the minimization of a functional. Therefore, methods for solving these problems typically require differentiating maps whose input and/or output are functions -- commonly referred to as variational differentiation. Such maps are not addressed at the mathematical level by the chain rule, which underlies modern symbolic and algorithmic differentiation (AD) systems. Although there are algorithmic solutions such as tracing and reverse accumulation, they do not provide human readability and introduce strict programming constraints that bottleneck performance, especially in high-performance computing (HPC) environments. In this manuscript, we propose a new computer theoretic model of differentiation by combining the pullback of the $\mathbf{B}$ and $\mathbf{C}$ combinators from the combinatory logic. Unlike frameworks based on the chain rule, this model differentiates a minimal complete basis for the space of computable functions. Consequently, the model is capable of analytic backpropagation and variational differentiation while supporting complex numbers. To demonstrate the generality of this approach we build a system named CombDiff, which can differentiate nontrivial variational problems such as Hartree-Fock (HF) theory and multilayer perceptrons. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_16154 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Automating Variational Differentiation Li, Kangbo Damle, Anil Mathematical Software Numerical Analysis Programming Languages 68V99, 49-04, 15-04 Many problems in Physics and Chemistry are formulated as the minimization of a functional. Therefore, methods for solving these problems typically require differentiating maps whose input and/or output are functions -- commonly referred to as variational differentiation. Such maps are not addressed at the mathematical level by the chain rule, which underlies modern symbolic and algorithmic differentiation (AD) systems. Although there are algorithmic solutions such as tracing and reverse accumulation, they do not provide human readability and introduce strict programming constraints that bottleneck performance, especially in high-performance computing (HPC) environments. In this manuscript, we propose a new computer theoretic model of differentiation by combining the pullback of the $\mathbf{B}$ and $\mathbf{C}$ combinators from the combinatory logic. Unlike frameworks based on the chain rule, this model differentiates a minimal complete basis for the space of computable functions. Consequently, the model is capable of analytic backpropagation and variational differentiation while supporting complex numbers. To demonstrate the generality of this approach we build a system named CombDiff, which can differentiate nontrivial variational problems such as Hartree-Fock (HF) theory and multilayer perceptrons. |
| title | Automating Variational Differentiation |
| topic | Mathematical Software Numerical Analysis Programming Languages 68V99, 49-04, 15-04 |
| url | https://arxiv.org/abs/2406.16154 |