An Introduction to Kernel and Operator Learning Methods for Homogenization by Self-consistent Clustering Analysis Article Swipe
Owen Huang
,
Sourav Saha
,
Jiachen Guo
,
Wing Kam Liu
·
YOU?
·
· 2022
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2212.00802
YOU?
·
· 2022
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2212.00802
Recent advances in operator learning theory have improved our knowledge about learning maps between infinite dimensional spaces. However, for large-scale engineering problems such as concurrent multiscale simulation for mechanical properties, the training cost for the current operator learning methods is very high. The article presents a thorough analysis on the mathematical underpinnings of the operator learning paradigm and proposes a kernel learning method that maps between function spaces. We first provide a survey of modern kernel and operator learning theory, as well as discuss recent results and open problems. From there, the article presents an algorithm to how we can analytically approximate the piecewise constant functions on R for operator learning. This implies the potential feasibility of success of neural operators on clustered functions. Finally, a k-means clustered domain on the basis of a mechanistic response is considered and the Lippmann-Schwinger equation for micro-mechanical homogenization is solved. The article briefly discusses the mathematics of previous kernel learning methods and some preliminary results with those methods. The proposed kernel operator learning method uses graph kernel networks to come up with a mechanistic reduced order method for multiscale homogenization.
Related Topics
Concepts
Operator (biology)
Kernel (algebra)
Computer science
Cluster analysis
Artificial intelligence
Kernel method
Piecewise
Homogenization (climate)
Mathematics
Algorithm
Machine learning
Applied mathematics
Support vector machine
Mathematical analysis
Discrete mathematics
Biodiversity
Ecology
Repressor
Chemistry
Gene
Biochemistry
Biology
Transcription factor
Metadata
- Type
- preprint
- Language
- en
- Landing Page
- http://arxiv.org/abs/2212.00802
- https://arxiv.org/pdf/2212.00802
- OA Status
- green
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W4310742953
All OpenAlex metadata
Raw OpenAlex JSON
- OpenAlex ID
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https://openalex.org/W4310742953Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.48550/arxiv.2212.00802Digital Object Identifier
- Title
-
An Introduction to Kernel and Operator Learning Methods for Homogenization by Self-consistent Clustering AnalysisWork title
- Type
-
preprintOpenAlex work type
- Language
-
enPrimary language
- Publication year
-
2022Year of publication
- Publication date
-
2022-12-01Full publication date if available
- Authors
-
Owen Huang, Sourav Saha, Jiachen Guo, Wing Kam LiuList of authors in order
- Landing page
-
https://arxiv.org/abs/2212.00802Publisher landing page
- PDF URL
-
https://arxiv.org/pdf/2212.00802Direct link to full text PDF
- Open access
-
YesWhether a free full text is available
- OA status
-
greenOpen access status per OpenAlex
- OA URL
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https://arxiv.org/pdf/2212.00802Direct OA link when available
- Concepts
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Operator (biology), Kernel (algebra), Computer science, Cluster analysis, Artificial intelligence, Kernel method, Piecewise, Homogenization (climate), Mathematics, Algorithm, Machine learning, Applied mathematics, Support vector machine, Mathematical analysis, Discrete mathematics, Biodiversity, Ecology, Repressor, Chemistry, Gene, Biochemistry, Biology, Transcription factorTop concepts (fields/topics) attached by OpenAlex
- Cited by
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0Total citation count in OpenAlex
- Related works (count)
-
10Other works algorithmically related by OpenAlex
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| abstract_inverted_index.theory | 5 |
| abstract_inverted_index.there, | 90 |
| abstract_inverted_index.article | 43, 92, 148 |
| abstract_inverted_index.between | 13, 65 |
| abstract_inverted_index.briefly | 149 |
| abstract_inverted_index.current | 35 |
| abstract_inverted_index.discuss | 83 |
| abstract_inverted_index.implies | 112 |
| abstract_inverted_index.k-means | 126 |
| abstract_inverted_index.methods | 38, 157 |
| abstract_inverted_index.provide | 70 |
| abstract_inverted_index.reduced | 181 |
| abstract_inverted_index.results | 85, 161 |
| abstract_inverted_index.solved. | 146 |
| abstract_inverted_index.spaces. | 16, 67 |
| abstract_inverted_index.success | 117 |
| abstract_inverted_index.theory, | 79 |
| abstract_inverted_index.Finally, | 124 |
| abstract_inverted_index.However, | 17 |
| abstract_inverted_index.advances | 1 |
| abstract_inverted_index.analysis | 47 |
| abstract_inverted_index.constant | 104 |
| abstract_inverted_index.equation | 141 |
| abstract_inverted_index.function | 66 |
| abstract_inverted_index.improved | 7 |
| abstract_inverted_index.infinite | 14 |
| abstract_inverted_index.learning | 4, 11, 37, 55, 61, 78, 156, 169 |
| abstract_inverted_index.methods. | 164 |
| abstract_inverted_index.networks | 174 |
| abstract_inverted_index.operator | 3, 36, 54, 77, 109, 168 |
| abstract_inverted_index.paradigm | 56 |
| abstract_inverted_index.presents | 44, 93 |
| abstract_inverted_index.previous | 154 |
| abstract_inverted_index.problems | 21 |
| abstract_inverted_index.proposed | 166 |
| abstract_inverted_index.proposes | 58 |
| abstract_inverted_index.response | 135 |
| abstract_inverted_index.thorough | 46 |
| abstract_inverted_index.training | 31 |
| abstract_inverted_index.algorithm | 95 |
| abstract_inverted_index.clustered | 122, 127 |
| abstract_inverted_index.discusses | 150 |
| abstract_inverted_index.functions | 105 |
| abstract_inverted_index.knowledge | 9 |
| abstract_inverted_index.learning. | 110 |
| abstract_inverted_index.operators | 120 |
| abstract_inverted_index.piecewise | 103 |
| abstract_inverted_index.potential | 114 |
| abstract_inverted_index.problems. | 88 |
| abstract_inverted_index.concurrent | 24 |
| abstract_inverted_index.considered | 137 |
| abstract_inverted_index.functions. | 123 |
| abstract_inverted_index.mechanical | 28 |
| abstract_inverted_index.multiscale | 25, 185 |
| abstract_inverted_index.simulation | 26 |
| abstract_inverted_index.approximate | 101 |
| abstract_inverted_index.dimensional | 15 |
| abstract_inverted_index.engineering | 20 |
| abstract_inverted_index.feasibility | 115 |
| abstract_inverted_index.large-scale | 19 |
| abstract_inverted_index.mathematics | 152 |
| abstract_inverted_index.mechanistic | 134, 180 |
| abstract_inverted_index.preliminary | 160 |
| abstract_inverted_index.properties, | 29 |
| abstract_inverted_index.analytically | 100 |
| abstract_inverted_index.mathematical | 50 |
| abstract_inverted_index.underpinnings | 51 |
| abstract_inverted_index.homogenization | 144 |
| abstract_inverted_index.homogenization. | 186 |
| abstract_inverted_index.micro-mechanical | 143 |
| abstract_inverted_index.Lippmann-Schwinger | 140 |
| cited_by_percentile_year | |
| countries_distinct_count | 0 |
| institutions_distinct_count | 4 |
| citation_normalized_percentile |