Understanding Loss Landscapes of Neural Network Models in Solving Partial Differential Equations. Article Swipe
Solving partial differential equations (PDEs) by parametrizing its solution by neural networks (NNs) has been popular in the past a few years. However, different types of loss functions can be proposed for the same PDE. For the Poisson equation, the loss function can be based on the weak formulation of energy variation or the least squares method, which leads to the deep Ritz model and deep Galerkin model, respectively. But loss landscapes from these different models give arise to different practical performance of training the NN parameters. To investigate and understand such practical differences, we propose to compare the loss landscapes of these models, which are both high dimensional and highly non-convex. In such settings, the is more important than the traditional eigenvalue analysis to describe the non-convexity. We contribute to the landscape comparisons by proposing a index to scientifically and quantitatively describe the heuristic concept of roughness of landscape around minimizers. This index is based on random projections and the variance of (normalized) total variation for one dimensional projected functions, and it is efficient to compute. A large index hints an oscillatory landscape profile as a severe challenge for the first order optimization method. We apply this index to the two models for the Poisson equation and our empirical results reveal a consistent general observation that the landscapes from the deep Galerkin method around its local minimizers are less rough than the deep Ritz method, which supports the observed gain in accuracy of the deep Galerkin method.
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- Type
- preprint
- Language
- en
- Landing Page
- http://export.arxiv.org/pdf/2103.11069
- OA Status
- green
- References
- 23
- Related Works
- 20
- OpenAlex ID
- https://openalex.org/W3136604563
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- OpenAlex ID
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https://openalex.org/W3136604563Canonical identifier for this work in OpenAlex
- Title
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Understanding Loss Landscapes of Neural Network Models in Solving Partial Differential Equations.Work title
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preprintOpenAlex work type
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enPrimary language
- Publication year
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2021Year of publication
- Publication date
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2021-03-20Full publication date if available
- Authors
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Keke Wu, Rui Du, Jingrun Chen, Xiang ZhouList of authors in order
- Landing page
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https://export.arxiv.org/pdf/2103.11069Publisher landing page
- Open access
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YesWhether a free full text is available
- OA status
-
greenOpen access status per OpenAlex
- OA URL
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https://export.arxiv.org/pdf/2103.11069Direct OA link when available
- Concepts
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Applied mathematics, Partial differential equation, Mathematics, Galerkin method, Mathematical optimization, Poisson's equation, Artificial neural network, Heuristic, Computer science, Poisson distribution, Mathematical analysis, Finite element method, Artificial intelligence, Statistics, Thermodynamics, PhysicsTop concepts (fields/topics) attached by OpenAlex
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0Total citation count in OpenAlex
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20Other works algorithmically related by OpenAlex
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| abstract_inverted_index.random | 159 |
| abstract_inverted_index.reveal | 214 |
| abstract_inverted_index.severe | 190 |
| abstract_inverted_index.years. | 21 |
| abstract_inverted_index.Poisson | 37, 208 |
| abstract_inverted_index.Solving | 0 |
| abstract_inverted_index.compare | 97 |
| abstract_inverted_index.concept | 147 |
| abstract_inverted_index.general | 217 |
| abstract_inverted_index.method, | 56, 238 |
| abstract_inverted_index.method. | 197, 250 |
| abstract_inverted_index.models, | 103 |
| abstract_inverted_index.partial | 1 |
| abstract_inverted_index.popular | 15 |
| abstract_inverted_index.profile | 187 |
| abstract_inverted_index.propose | 95 |
| abstract_inverted_index.results | 213 |
| abstract_inverted_index.squares | 55 |
| abstract_inverted_index.Galerkin | 66, 225, 249 |
| abstract_inverted_index.However, | 22 |
| abstract_inverted_index.accuracy | 245 |
| abstract_inverted_index.analysis | 124 |
| abstract_inverted_index.compute. | 178 |
| abstract_inverted_index.describe | 126, 144 |
| abstract_inverted_index.equation | 209 |
| abstract_inverted_index.function | 41 |
| abstract_inverted_index.networks | 11 |
| abstract_inverted_index.observed | 242 |
| abstract_inverted_index.proposed | 30 |
| abstract_inverted_index.solution | 8 |
| abstract_inverted_index.supports | 240 |
| abstract_inverted_index.training | 83 |
| abstract_inverted_index.variance | 163 |
| abstract_inverted_index.challenge | 191 |
| abstract_inverted_index.different | 23, 74, 79 |
| abstract_inverted_index.efficient | 176 |
| abstract_inverted_index.empirical | 212 |
| abstract_inverted_index.equation, | 38 |
| abstract_inverted_index.equations | 3 |
| abstract_inverted_index.functions | 27 |
| abstract_inverted_index.heuristic | 146 |
| abstract_inverted_index.important | 119 |
| abstract_inverted_index.landscape | 133, 151, 186 |
| abstract_inverted_index.practical | 80, 92 |
| abstract_inverted_index.projected | 171 |
| abstract_inverted_index.proposing | 136 |
| abstract_inverted_index.roughness | 149 |
| abstract_inverted_index.settings, | 114 |
| abstract_inverted_index.variation | 51, 167 |
| abstract_inverted_index.consistent | 216 |
| abstract_inverted_index.contribute | 130 |
| abstract_inverted_index.eigenvalue | 123 |
| abstract_inverted_index.functions, | 172 |
| abstract_inverted_index.landscapes | 71, 100, 221 |
| abstract_inverted_index.minimizers | 230 |
| abstract_inverted_index.understand | 90 |
| abstract_inverted_index.comparisons | 134 |
| abstract_inverted_index.dimensional | 108, 170 |
| abstract_inverted_index.formulation | 48 |
| abstract_inverted_index.investigate | 88 |
| abstract_inverted_index.minimizers. | 153 |
| abstract_inverted_index.non-convex. | 111 |
| abstract_inverted_index.observation | 218 |
| abstract_inverted_index.oscillatory | 185 |
| abstract_inverted_index.parameters. | 86 |
| abstract_inverted_index.performance | 81 |
| abstract_inverted_index.projections | 160 |
| abstract_inverted_index.traditional | 122 |
| abstract_inverted_index.(normalized) | 165 |
| abstract_inverted_index.differences, | 93 |
| abstract_inverted_index.differential | 2 |
| abstract_inverted_index.optimization | 196 |
| abstract_inverted_index.parametrizing | 6 |
| abstract_inverted_index.respectively. | 68 |
| abstract_inverted_index.non-convexity. | 128 |
| abstract_inverted_index.quantitatively | 143 |
| abstract_inverted_index.scientifically | 141 |
| cited_by_percentile_year | |
| countries_distinct_count | 1 |
| institutions_distinct_count | 4 |
| citation_normalized_percentile |