Predictive models of thermodynamic properties of mixtures are paramount in chemical engineering and chemistry. Classical thermodynamic models are successful in generalizing over (continuous) conditions like temperature and concentration. On the other hand, matrix completion methods (MCMs) from machine learning successfully generalize over (discrete) binary systems; these MCMs can make predictions without any data for a given binary system by implicitly learning commonalities across systems. In the present work, we combine the strengths from both worlds in a hybrid approach. The underlying idea is to predict the pairinteraction energies , as they are used in basically all physical models of liquid mixtures, by an MCM. As an example, we embed an MCM into UNIQUAC, a widelyused physical model for the Gibbs excess energy. We train the resulting hybrid model in a Bayesian machinelearning framework on experimental data for activity coefficients in binary systems of 1146 components from the Dortmund Data Bank. We thereby obtain, for the first time, a complete set of UNIQUAC parameters for all binary systems of these components, which allows us to predict, in principle, activity coefficients at arbitrary temperature and composition for any combination of these components, not only for binary but also for multicomponent systems. The hybrid model even outperforms the best available physical model for predicting activity coefficients, the modified UNIFAC (Dortmund) model.
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A framework for machine learning of model error in dynamical systems
The development of datainformed predictive models for dynamical systems is of widespread interest in many disciplines. We present a unifying framework for blending mechanistic and machinelearning approaches to identify dynamical systems from noisily and partially observed data. We compare pure datadriven learning with hybrid models which incorporate imperfect domain knowledge, referring to the discrepancy between an assumed truth model and the imperfect mechanistic model as model error. Our formulation is agnostic to the chosen machine learning model, is presented in both continuous and discretetime settings, and is compatible both with model errors that exhibit substantial memory and errors that are memoryless. First, we study memoryless linear (w.r.t. parametricdependence) model error from a learning theory perspective, defining excess risk and generalization error. For ergodic continuoustime systems, we prove that both excess risk and generalization error are bounded above by terms that diminish with the squareroot of T T , the timeinterval over which training data is specified. Secondly, we study scenarios that benefit from modeling with memory, proving universal approximation theorems for two classes of continuoustime recurrent neural networks (RNNs): both can learn memorydependent model error, assuming that it is governed by a finitedimensional hidden variable and that, together, the observed and hidden variables form a continuoustime Markovian system. In addition, we connect one class of RNNs to reservoir computing, thereby relating learning of memorydependent error to recent work on supervised learning between Banach spaces using random features. Numerical results are presented (Lorenz ’63, Lorenz ’96 Multiscale systems) to compare purely datadriven and hybrid approaches, finding hybrid methods less datahungry and more parametrically efficient. We also find that, while a continuoustime framing allows for robustness to irregular sampling and desirable domain interpretability, a discretetime framing can provide similar or better predictive performance, especially when data are undersampled and the vector field defining the true dynamics cannot be identified. Finally, we demonstrate numerically how data assimilation can be leveraged to learn hidden dynamics from noisy, partiallyobserved data, and illustrate challenges in representing memory by this approach, and in the training of such models.
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 Award ID(s):
 1835860
 NSFPAR ID:
 10377117
 Date Published:
 Journal Name:
 Communications of the American Mathematical Society
 Volume:
 2
 Issue:
 7
 ISSN:
 26923688
 Page Range / eLocation ID:
 283 to 344
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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