Dual neural network architecture for determining epistemic and aleatoric uncertainties

1. A neural network system comprising:

a first neural network configured to predict a mean value output and epistemic uncertainty of the output given input data;

a second neural network configured to predict total uncertainty of the output of the first neural network, wherein the second neural network is trained to predict the total uncertainty of the output of the first neural network given the input data through a training process involving minimizing a cost function that involves differences between a predicted mean value of a geophysical property of a geological formation from the first neural network and a ground-truth value of the geophysical property of the geological formation; and

one or more processors configured to run a software module that determines aleatoric uncertainty of the output of the first neural network based on the epistemic uncertainty of the output and the total uncertainty of the output.

2. The neural network system of claim 1 , wherein:

the first neural network comprises a Bayesian neural network or an artificial neural network with dropout.

3. The neural network system of claim 1 , wherein:

the second neural network comprises an artificial neural network or a Bayesian neural network.

4. The neural network system of claim 1 , wherein:

the aleatoric uncertainty of the output of the first neural network is sensitive to heteroscedastic noise in the input data.

5. The neural network system of claim 1 , wherein:

the first neural network is trained by minimizing a corresponding cost function that involves a corresponding ground-truth value of the output.

6. The neural network system of claim 1 , wherein:

error of the first neural network is used to train the second neural network to learn a mapping between the input data and the total uncertainty of the output of the first neural network.

7. The neural network system of claim 1 , wherein:

the input data is applied to both the first neural network and the second neural network simultaneously or in a parallel manner.

8. The neural network system of claim 1 , wherein:

the first neural network is trained to predict a mean value and epistemic uncertainty of an unknown geophysical property of the geological formation given a vector of measurement data of the geological formation as input;

the second neural network is trained to predict total uncertainty of the unknown geophysical property of the geological formation given the vector of measurement data of the geological formation as input; and

the software module is configured to determine aleatoric uncertainty of the unknown geophysical property of the geological formation based on the epistemic uncertainty of the unknown geophysical property of the geological formation and the total uncertainty of the unknown geophysical property of the geological formation.

9. The neural network system of claim 1 , wherein:

at least one of the first neural network or the second neural network is realized by the one or more processors.

10. A machine learning method comprising:

in a training phase, training a first neural network to predict an output and epistemic uncertainty of the output given input data, and training a second neural network to predict total uncertainty of the output of the first neural network, wherein the second neural network is trained to predict the total uncertainty of the output of the first neural network given the input data through a training process involving minimizing a cost function that involves differences between a predicted mean value of a geophysical property of a geological formation from the first neural network and a ground-truth value of the geophysical property of the geological formation; and

in an inference phase, supplying input data to the trained first neural network to predict an output and epistemic uncertainty of the output given the input data, using the trained second neural network to predict total uncertainty of the output of the trained first neural network, and determining aleatoric uncertainty of the output of the first neural network based on the epistemic uncertainty of the output and the total uncertainty of the output.

11. The method of claim 10 , wherein:

the training phase further comprises using error of the first neural network to train the second neural network to learn a mapping between the input data and the total uncertainty of the output of the first neural network.

12. The method of claim 10 , wherein:

the training phase further comprises applying the input data to both the first neural network and the second neural network simultaneously or in a parallel manner.

13. The method of claim 10 , wherein:

the first neural network comprises a Bayesian neural network or an artificial neural network with dropout.

14. The method of claim 10 , wherein:

the second neural network comprises an artificial neural network or a Bayesian neural network.

15. The method of claim 10 , wherein:

the aleatoric uncertainty of the output of the first neural network is sensitive to heteroscedastic noise in the input data.

16. The method of claim 10 , wherein:

the training phase further comprises training the first neural network by minimizing a corresponding cost function that involves a corresponding ground-truth value of the output.

17. A machine learning method for geological formation modeling comprising:

in a training phase, training a first neural network to predict a mean value and epistemic uncertainty of an unknown geophysical property of a geological formation given a vector of measurement data of the geological formation as input, and training a second neural network to predict total uncertainty of the unknown geophysical property of the geological formation given the vector of measurement data of the geological formation as input, wherein the training phase further comprises training the second neural network by minimizing a cost function that involves differences between a predicted mean value of the unknown geophysical property of the geological formation from the first neural network and a ground-truth value of the unknown geophysical property of the geological formation;

in an inference phase, supplying at least one vector of measurement data of the geological formation to the trained first neural network to predict a mean value and epistemic uncertainty of the unknown geophysical property of the geological formation, using the trained second neural network to predict total uncertainty of the unknown geophysical property of the geological formation, and determining aleatoric uncertainty of the unknown geophysical property of the geological formation based on the epistemic uncertainty and the total uncertainty of the unknown geophysical property of the geological formation.

18. The method of claim 17 , wherein:

the aleatoric uncertainty of the unknown geophysical property of the geological formation is sensitive to heteroscedastic noise in the measurement data of the geological formation.

19. The method of claim 17 , wherein:

the first neural network comprises a Bayesian neural network or an artificial neural network with dropout.

20. The method of claim 17 , wherein:

the second neural network comprises an artificial neural network or a Bayesian neural network.

21. The method of claim 17 , wherein:

the training phase further comprises training the first neural network by minimizing a corresponding cost function that involves a corresponding ground-truth value of the unknown geophysical property of the geological formation.

22. The method of claim 17 , wherein:

at least one of the first neural network and the second neural network is realized by a processor.

23. The method of claim 17 , wherein:

the training phase further comprises applying the vector of measurement data of the geological formation to both the first neural network and the second neural network simultaneously or in a parallel manner.

24. The method of claim 17 , further comprising:

performing the operations of the inference phase are performed over a plurality of vectors of measurement data of the geological formation at varying depths to predict the geophysical property of the geological formation and associated epistemic uncertainty and aleatoric uncertainty at the varying depths.