Generalizing Deep Learning Models for Varied Diffusion Equations

21 Jun 2024


(1) J. Quetzalcóatl Toledo-Marín, University of British Columbia, BC Children’s Hospital Research Institute Vancouver BC, Canada (Email:;

(2) James A. Glazier, Biocomplexity Institute and Department of Intelligent Systems Engineering, Indiana University, Bloomington, IN 47408, USA (Email:;

(3) Geoffrey Fox, University of Virginia, Computer Science and Biocomplexity Institute, 994 Research Park Blvd, Charlottesville, Virginia, 22911, USA (Email:

Abstract and Introduction




Conclusions and References

5 Conclusions

When selecting a NN for a specific task, it is important to consider the function and requirements of the task at hand. There is currently no consensus on which NN is optimal for a given task, primarily due to the large number of NN options available, the rapidly evolving nature of the field, and the lack of a comprehensive deep learning theory. This leads to a reliance on empirical results. Our paper is an important step at establishing best practices for this type of problems. we focused on randomly placed sources with random fluxes which yield large variations in the field. Our method can be generalized for different diffusion equations.

As part of future steps, we will increase the complexity of the problem being solved by considering conditions closer to real-problems, i.e., by considering less symmetrical sources, different diffusivities and different boundary conditions. In addition, further design is required for these models to be used in a production environment in a reliable way, i.e., how to deal with error performance edge cases on-the-fly? Ultimately, to be able to deploy the model for production, one requires a method to keep the performance error below a predefined bound. For instance, one can train an additional NN that takes the predicted stationary solution and predicts the initial condition, which is then compared with the ground truth initial condition. This framework allows a comparison between ground truth input and predicted input without requiring the ground truth steady-state solution. However, this approach would only be reliable if the input error is always proportional to the steady-state error and hence requires further investigation. A perhaps simpler approach consists in sampling the NN’s output for the same input using a drop-out feature, such that if the fluctuations from the samples are small enough, then the NN’s prediction is robust enough. Both cases require benchmark design. We have developed a process in this paper that can easily be replicated for more complicated problems of this type, and provided a variety of benchmarks.

6 Acknowledgements

JQTM acknowledges a Mitacs Postdoctoral Fellowship. GCF acknowledges partial support from DOE DE-SC0023452 and NSF OAC-2204115.


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