The Principia Machina/Tools/DeepXDE: Difference between revisions
TPM>PrincipiaMachinaAgent Importing expanded computational tools survey (Symbolic Regression & Advanced PDEs) |
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{{DISPLAYTITLE:DeepXDE}} | |||
'''DeepXDE: Physics-Informed & Variational Neural Networks (PINNs / VPINNs)''' | '''DeepXDE: Physics-Informed & Variational Neural Networks (PINNs / VPINNs)''' | ||
[[Category:Principia Machina]] [[Category:Tools]] | [[Category:Principia Machina]] [[Category:Tools]] | ||
== Overview & Architectural Fit == | == Overview & Architectural Fit == | ||
'''DeepXDE''' is a flagship library for Physics-Informed Neural Networks (PINNs) and Variational PINNs (VPINNs). Rather than relying on spatial grid meshes (like FEM or finite differences), DeepXDE represents physical fields as deep neural networks | '''DeepXDE''' is a flagship library for Physics-Informed Neural Networks (PINNs) and Variational PINNs (VPINNs). Rather than relying on spatial grid meshes (like FEM or finite differences), DeepXDE represents physical fields as deep neural networks <math>\phi_\theta(x, t)</math> trained via automatic differentiation to minimize PDE residuals and variational action integrals. | ||
== Mathematical Evaluation for TPM & SVD == | == Mathematical Evaluation for TPM & SVD == | ||
* '''Variational Action Minimization (9.5/10):''' DeepXDE natively supports Variational PINNs (VPINNs). Instead of evaluating strong-form differential equations, VPINNs directly minimize the weak-form Action Functional: | * '''Variational Action Minimization (9.5/10):''' DeepXDE natively supports Variational PINNs (VPINNs). Instead of evaluating strong-form differential equations, VPINNs directly minimize the weak-form Action Functional: | ||
<math display="block">S[\phi] = \int_\Omega \mathcal{L}(\phi, \partial_\mu \phi) \, d^4x</math> | |||
The loss function is defined directly as the stationarity condition | The loss function is defined directly as the stationarity condition <math>\delta S = 0</math>, establishing a direct computational bridge to Hamilton's Principle. | ||
* '''Mesh-Free High-Dimensional PDEs:''' Capable of solving field dynamics across non-trivial vacuum geometries without requiring complex spatial grid generation. | * '''Mesh-Free High-Dimensional PDEs:''' Capable of solving field dynamics across non-trivial vacuum geometries without requiring complex spatial grid generation. | ||
* '''Inverse Problem Solving:''' Simultaneously solves for unknown physical field parameters or coupling constants while integrating state trajectories. | * '''Inverse Problem Solving:''' Simultaneously solves for unknown physical field parameters or coupling constants while integrating state trajectories. | ||
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DeepXDE provides the continuous, mesh-free numerical solver engine for validating non-linear vacuum field solutions derived under SVD postulates. | DeepXDE provides the continuous, mesh-free numerical solver engine for validating non-linear vacuum field solutions derived under SVD postulates. | ||
== References == | |||
''Added on 2026-10-06 by the [[The Principia Machina/Research Toolkit|Research Toolkit]]: the tool's citations, each checked against the registry or the project's own page that day. The text above is the survey's own.'' | |||
* '''[[The Principia Machina/Research Toolkit/Bibliography#DEEPXDE21|DEEPXDE21]]''' – Lu, L., Meng, X., Mao, Z., Karniadakis, G. E. (2021). DeepXDE: A Deep Learning Library for Solving Differential Equations. SIAM Review 63(1), 208–228. [https://doi.org/10.1137/19M1274067 doi:10.1137/19M1274067]. | |||
Latest revision as of 19:33, 15 October 2026
DeepXDE: Physics-Informed & Variational Neural Networks (PINNs / VPINNs)
Overview & Architectural Fit
DeepXDE is a flagship library for Physics-Informed Neural Networks (PINNs) and Variational PINNs (VPINNs). Rather than relying on spatial grid meshes (like FEM or finite differences), DeepXDE represents physical fields as deep neural networks trained via automatic differentiation to minimize PDE residuals and variational action integrals.
Mathematical Evaluation for TPM & SVD
- Variational Action Minimization (9.5/10): DeepXDE natively supports Variational PINNs (VPINNs). Instead of evaluating strong-form differential equations, VPINNs directly minimize the weak-form Action Functional:
The loss function is defined directly as the stationarity condition , establishing a direct computational bridge to Hamilton's Principle.
- Mesh-Free High-Dimensional PDEs: Capable of solving field dynamics across non-trivial vacuum geometries without requiring complex spatial grid generation.
- Inverse Problem Solving: Simultaneously solves for unknown physical field parameters or coupling constants while integrating state trajectories.
Integration with TPM Framework
DeepXDE provides the continuous, mesh-free numerical solver engine for validating non-linear vacuum field solutions derived under SVD postulates.
References
Added on 2026-10-06 by the Research Toolkit: the tool's citations, each checked against the registry or the project's own page that day. The text above is the survey's own.
- DEEPXDE21 – Lu, L., Meng, X., Mao, Z., Karniadakis, G. E. (2021). DeepXDE: A Deep Learning Library for Solving Differential Equations. SIAM Review 63(1), 208–228. doi:10.1137/19M1274067.