Inverse modeling of porous flow through deep neural networks: the case of coffee percolation

TL;DR

This paper develops a mathematically grounded deep learning approach to solve the inverse problem of coffee extraction, enabling the reconstruction of brewing parameters from chemical profiles with high accuracy.

Contribution

It introduces a novel inverse modeling framework combining multiphysics simulations, analytical conditions, and deep learning for personalized coffee brewing optimization.

Findings

Accurately reconstructs brewing temperature, grind size, and powder composition.

Provides a rigorous mathematical foundation for the inverse problem.

Demonstrates effectiveness through extensive validation experiments.

Abstract

This work addresses the inverse problem of espresso coffee extraction, in which one aims to reconstruct the brewing conditions that generate a desired chemical profile in the final beverage. Starting from a high-fidelity multiphysics percolation model, describing fluid flow, solute transport, solid, liquid reactions, and heat exchange within the coffee bed, we derive a reduced forward operator mapping controllable brewing parameters to the concentrations of the main chemical species in the cup. From a mathematical standpoint, we formalize the structural requirements for the local solvability of inverse problems, providing a minimal analytical condition for the existence of a (local) inverse map: continuous differentiability of the forward operator and a locally constant, nondegenerate Jacobian rank. Under these assumptions, the Constant Rank Theorem ensures that the image of the forward operator is a smooth embedded manifold on which well-defined local right-inverses exist. Extensive experiments, including off-grid validation, show that the learned inverse map accurately reconstructs brewing temperature, grind size, and powder composition. The resulting framework combines rigorous analytical guarantees with modern data-driven methods, providing a principled and computationally efficient solution to the inverse extraction problem and enabling personalised brewing, recipe optimisation, and integration into smart coffee-machine systems.