Addressing discontinuous root-finding for subsequent differentiability in machine learning, inverse problems, and control
Abstract/Contents
- Abstract
- There are many physical processes that have inherent discontinuities in their mathematical formulations. This thesis is motivated by the specific case of collisions between two rigid or deformable bodies and the intrinsic nature of that discontinuity. The impulse response to a collision is discontinuous with the lack of any response when no collision occurs, which causes difficulties for numerical approaches that require differentiability which are typical in machine learning, inverse problems, and control. We theoretically and numerically demonstrate that the derivative of the collision time with respect to the parameters becomes infinite as one approaches the barrier separating colliding from not colliding, and use lifting to complexify the solution space so that solutions on the other side of the barrier are directly attainable as precise values. Subsequently, we mollify the barrier posed by the unbounded derivatives, so that one can tunnel back and forth in a smooth and reliable fashion facilitating the use of standard numerical approaches. Moreover, we illustrate that standard approaches fail in numerous ways mostly due to a lack of understanding of the mathematical nature of the problem (e.g. typical backpropagation utilizes many rules of differentiation, but ignores L'Hopital's rule).
Description
| Type of resource | text |
|---|---|
| Form | electronic resource; remote; computer; online resource |
| Extent | 1 online resource. |
| Place | California |
| Place | [Stanford, California] |
| Publisher | [Stanford University] |
| Copyright date | 2023; ©2023 |
| Publication date | 2023; 2023 |
| Issuance | monographic |
| Language | English |
Creators/Contributors
| Author | Johnson, Daniel Scott |
|---|---|
| Degree supervisor | Fedkiw, Ronald P, 1968- |
| Thesis advisor | Fedkiw, Ronald P, 1968- |
| Thesis advisor | Farhat, Charbel |
| Thesis advisor | Liu, Cheng-Yun Karen, 1977- |
| Degree committee member | Farhat, Charbel |
| Degree committee member | Liu, Cheng-Yun Karen, 1977- |
| Associated with | Stanford University, School of Engineering |
| Associated with | Stanford University, Computer Science Department |
Subjects
| Genre | Theses |
|---|---|
| Genre | Text |
Bibliographic information
| Statement of responsibility | Daniel Scott Johnson. |
|---|---|
| Note | Submitted to the Computer Science Department. |
| Thesis | Thesis Ph.D. Stanford University 2023. |
| Location | https://purl.stanford.edu/gh289mn8096 |
Access conditions
- Copyright
- © 2023 by Daniel Scott Johnson
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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