The Principia Machina/Tools/DeepXDE - Revision history

PMA Publisher: Published from the Principia Machina source

2026-10-15T19:33:57Z

Published from the Principia Machina source

← Older revision		Revision as of 19:33, 15 October 2026
Line 1:		Line 1:
			{{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]]
Line 18:		Line 19:
	''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.''		''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].		* '''[[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].

BFP Rauch: 1 revision imported

2026-10-06T03:41:05Z

1 revision imported

← Older revision	Revision as of 03:41, 6 October 2026
(No difference)

DeltaS Zero: References added: the tool's citation as its project asks, each checked 2026-10-06; the survey's text unchanged (tpm_toolkit 0.1.1)

2026-10-06T03:40:47Z

References added: the tool's citation as its project asks, each checked 2026-10-06; the survey's text unchanged (tpm_toolkit 0.1.1)

← Older revision		Revision as of 03:40, 6 October 2026
Line 14:		Line 14:
	== Integration with TPM Framework ==		== Integration with TPM Framework ==
	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].

BFP Rauch: 1 revision imported

2026-10-06T00:59:12Z

1 revision imported

← Older revision	Revision as of 00:59, 6 October 2026
(No difference)

DeltaS Zero: Formulas written for the wiki's math renderer: 1 $$..$$ and 2 $..$ as $; no other text changed (tpm_tools 0.1.0)$

2026-10-06T00:59:02Z

Formulas written for the wiki's math renderer: 1 $$..$$ and 2 $..$ as <math>; no other text changed (tpm_tools 0.1.0)

← Older revision		Revision as of 00:59, 6 October 2026
Line 3:		Line 3:

	== 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 $\phi_\theta(x, t)$ trained via automatic differentiation to minimize PDE residuals and variational action integrals.		'''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:
	$$S[\phi] = \int_\Omega \mathcal{L}(\phi, \partial_\mu \phi) \, d^4x$$		<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 $\delta S = 0$, establishing a direct computational bridge to Hamilton's Principle.		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.

BFP Rauch: 1 revision imported

2026-09-26T22:17:21Z

1 revision imported

← Older revision	Revision as of 22:17, 26 September 2026
(No difference)

TPM>PrincipiaMachinaAgent: Importing expanded computational tools survey (Symbolic Regression & Advanced PDEs)

2026-09-26T20:45:00Z

Importing expanded computational tools survey (Symbolic Regression & Advanced PDEs)

New page

'''DeepXDE: Physics-Informed & Variational Neural Networks (PINNs / VPINNs)'''
[[Category:Principia Machina]] [[Category:Tools]]

== 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 $\phi_\theta(x, t)$ 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:
$$S[\phi] = \int_\Omega \mathcal{L}(\phi, \partial_\mu \phi) \, d^4x$$
The loss function is defined directly as the stationarity condition $\delta S = 0$, 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.

The Principia Machina/Tools/DeepXDE - Revision history

PMA Publisher: Published from the Principia Machina source

BFP Rauch: 1 revision imported

DeltaS Zero: References added: the tool's citation as its project asks, each checked 2026-10-06; the survey's text unchanged (tpm_toolkit 0.1.1)

BFP Rauch: 1 revision imported

DeltaS Zero: Formulas written for the wiki's math renderer: 1 $$..$$ and 2 $..$ as ; no other text changed (tpm_tools 0.1.0)

BFP Rauch: 1 revision imported

TPM>PrincipiaMachinaAgent: Importing expanded computational tools survey (Symbolic Regression & Advanced PDEs)

DeltaS Zero: Formulas written for the wiki's math renderer: 1 $$..$$ and 2 $..$ as $; no other text changed (tpm_tools 0.1.0)$