Deep learning for partial differential equation (PDE) based models

1. A method for deep learning for partial differential equation (PDE)-based models by a processor, comprising:

generating a trained forecasting model and consistency constraints using a PDE-based model, a discretization of the PDE-based model, historical inputs of the PDE-based model, and a representation of consistency constraints to generate a predictive output, wherein the consistency constraints include consistency constraints from a higher resolution PDE-based model, consistency constraints from a lower resolution PDE-based model, and consistency constraints from adjacent tiles of a similar resolution PDE-based model.

2. The method of claim 1 , further including defining the consistency constraints as bounds on output values by the PDE-based model at selected points.

3. The method of claim 2 , further including:

acquiring the bounds from a measurement chain error of one or more sensor measurements; or

defining the bounds based on a total variance of across a patch of a mesh.

4. The method of claim 1 , further including:

defining the consistency constraints as bounds on a modulus of continuity;

defining the consistency constraints as bounds on one or more sub-gradients; or

defining the consistency constraints as bounds on a sum of the sub-gradients across all tiles of mesh.

5. The method of claim 1 , further including generating, from the trained forecasting model, additional inputs to use for training of additional forecasting models.

6. The method of claim 1 , further including:

applying data assimilation for training the trained forecasting model using a machine learning operation; or

performing on-line learning at run-time using the machine learning operation for training the trained forecasting model.

7. A system for deep learning for partial differential equation (PDE)-based models, comprising:

one or more computers with executable instructions that when executed cause the system to:

generate a trained forecasting model and consistency constraints using a PDE-based model, a discretization of the PDE-based model, historical inputs of the PDE-based model, and a representation of consistency constraints to generate a predictive output, wherein the consistency constraints include consistency constraints from a higher resolution PDE-based model, consistency constraints from a lower resolution PDE-based model, and consistency constraints from adjacent tiles of a similar resolution PDE-based model.

8. The system of claim 7 , wherein the executable instructions define the consistency constraints as bounds on output values by the PDE-based model at selected points.

9. The system of claim 8 , wherein the executable instructions:

acquire the bounds from a measurement chain error of one or more sensor measurements; or

defines the bounds based on a total variance of across a patch of a mesh.

10. The system of claim 7 , wherein the executable instructions:

define the consistency constraints as bounds on a modulus of continuity;

define the consistency constraints as bounds on one or more sub-gradients; or

define the consistency constraints as bounds on a sum of the sub-gradients across all tiles of mesh.

11. The system of claim 7 , wherein the executable instructions generate, from the trained forecasting model, additional inputs to use for training of additional forecasting models.

12. The system of claim 7 , wherein the executable instructions:

apply data assimilation for training the trained forecasting model using a machine learning operation; or

perform on-line learning at run-time using the machine learning operation for training the trained forecasting model.

13. A computer program product for, by a processor, deep learning for partial differential equation (PDE)-based models, the computer program product comprising a non-transitory computer-readable storage medium having computer-readable program code portions stored therein, the computer-readable program code portions comprising:

an executable portion that generates a trained forecasting model and consistency constraints using a PDE-based model, a discretization of the PDE-based model, historical inputs of the PDE-based model, and a representation of consistency constraints to generate a predictive output, wherein the consistency constraints include consistency constraints from a higher resolution PDE-based model, consistency constraints from a lower resolution PDE-based model, and consistency constraints from adjacent tiles of a similar resolution PDE-based model.

14. The computer program product of claim 13 , further including an executable portion that defines the consistency constraints as bounds on output values by the PDE-based model at selected points.

15. The computer program product of claim 14 , further including an executable portion that:

acquires the bounds from a measurement chain error of one or more sensor measurements; or

defines the bounds based on a total variance of across a patch of a mesh.

16. The computer program product of claim 13 , further including an executable portion that:

defines the consistency constraints as bounds on a modulus of continuity;

defines the consistency constraints as bounds on one or more sub-gradients; or

defines the consistency constraints as bounds on a sum of the sub-gradients across all tiles of mesh.

17. The computer program product of claim 13 , further including an executable portion that:

generate, from the trained forecasting model, additional inputs to use for training of additional forecasting models;

apply data assimilation for training the trained forecasting model using a machine learning operation; or

perform on-line learning at run-time using the machine learning operation for training the trained forecasting model.