(165e) An Implicit Mapping Approach for Process Systems Engineering Applications Using Automatic Differentiation and the Implicit Function Theorem

Performing input-output mapping is a challenging task in process systems engineering (PSE) applications, both in the forward and inverse directions. When dealing with the forward direction, it is of paramount importance to relate physical parameters via mathematical modeling, in which often the fundamental laws of physics are adequately understood [1]. Conversely, when dealing with the inverse mapping (or inverse problems in general), in which it is desired to obtain the input space (causes) given a set of observations (effects) defined in the output space, current approaches involve either exhaustive search in the parameters space or optimization techniques [2]. However, optimization-based approaches for solving the inverse mapping problem may be challenged by dimensionality [3].

To address this challenge, a unified framework for input-output mapping in either forward or inverse directions is proposed, in which the underlying process model is treated as an implicit function. Recent advances in differentiable programming [4] and automatic differentiation [5] allow use of the implicit function theorem and path integration to efficiently compute model solutions based on existing solutions in the neighborhood from domain to image (that will change from input to output space depending on the direction of the mapping). This framework can circumvent resorting to exhaustive search or nonlinear programming-based approaches for inverse mapping tasks.

Case studies related to PSE applications [6], particularly involving mapping sets in the input and output spaces for optimal operation of energy and chemical systems, are addressed to illustrate the effectiveness of the proposed framework. The obtained results are compared to typical mapping techniques, showing that the proposed approach is capable of finding the same solutions while the computational complexity is significantly reduced. This work is therefore a step forward towards addressing mapping tasks to obtain direct solutions using innovative PSE and numerical methods tools and techniques.

References

[1] R. C. Aster, B. Borchers, C. H. Thurber, “Parameter estimation and inverse problems”. (2018). doi:10.1016/C2015-0-02458-3.

[2] F. Ceccon, J. Jalving, J. Haddad, A. Thebelt, C. Tsay, C. D. Laird, R. Misener, Omlt: “Optimization & machine learning toolkit”. (2022). doi:arXiv:2202.02414.

[3] J. C. Carrasco, F. V. Lima, “Bilevel and parallel programing-based operability approaches for process intensification and modularity”. AIChE Journal 64 (8) 3042–3054 (2018). doi: 10.1002/aic.16113.

[4] A. G. Baydin, B. A. Pearlmutter, A. A. Radul, J. M. Siskind, “Automatic differentiation in machine learning: a survey”. Journal of Machine Learning Research 18 (153) (2018) 1–43. URL http://jmlr.org/papers/v18/17-468.html.

[5] J. Bradbury, R. Frostig, P. Hawkins, M. J. Johnson, C. Leary, D. Maclaurin, G. Necula, A. Paszke, J. VanderPlas, S. Wanderman-Milne, Q. Zhang, “JAX: composable transformations of Python+NumPy programs”. (2018). URL http://github.com/google/jax.

[6] V. Alves, J. R. Kitchin and F. V. Lima, “An Inverse Mapping Approach for Process Systems Engineering Applications Using Automatic Differentiation and the Implicit Function Theorem.” In Press (2023).