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Physics-informed Machine Learning has recently become attractive for learningphysical parameters and features from simulation and observation data. However,most existing methods do not ensure that the physics, such as balance laws(e.g., mass, momentum, energy conservation), are constrained. Some recent works(e.g., physics-informed neural networks) softly enforce physics constraints byincluding partial differential equation (PDE)-based loss functions but needre-discretization of the PDEs using auto-differentiation. Training these neuralnets on observational data showed that one could solve forward and inverseproblems in one shot. They evaluate the state variables and the parameters in aPDE. This re-discretization of PDEs is not necessarily an attractive option fordomain scientists that work with physics-based codes that have been developedfor decades with sophisticated discretization techniques to solve complexprocess models and advanced equations of state. This paper proposes a physicsconstrained machine learning framework, AdjointNet, allowing domain scientiststo embed their physics code in neural network training workflows. Thisembedding ensures that physics is constrained everywhere in the domain.Additionally, the mathematical properties such as consistency, stability, andconvergence vital to the numerical solution of a PDE are still satisfied. Weshow that the proposed AdjointNet framework can be used for parameterestimation (and uncertainty quantification by extension) and experimentaldesign using active learning. The applicability of our framework isdemonstrated for four flow cases. Results show that AdjointNet-based inversioncan estimate process model parameters with reasonable accuracy. These examplesdemonstrate the applicability of using existing software with no changes insource code to perform accurate and reliable inversion of model parameters.

AdjointNet: Constraining machine learning models with physics-based codes