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Hybrid models composing mechanistic ODE- based dynamics with flexible and expressive neural network components have grown rapidly in popularity, especially in scientific domains where such ODE-based modeling offers important interpretability and validated causal grounding (e.g., for counterfactual reasoning). The incorporation of mechanistic models also provides inductive bias in standard blackbox modeling approaches, critical when learning from small datasets or partially observed, complex systems. Unfortunately, as the hybrid models become more flexible, the causal grounding provided by the mechanistic model can quickly be lost. We address this problem by leveraging another common source of domain knowledge: ranking of treatment effects for a set of interventions, even if the precise treatment effect is unknown. We encode this information in a causal loss that we combine with the standard predictive loss to arrive at a hybrid loss that biases our learning towards causally valid hybrid models. We demonstrate our ability to achieve a win-win, state-of-the-art predictive performance and causal validity, in the challenging task of modeling glucose dynamics post-exercise in individuals with type 1 diabetes. 

The development of data-informed predictive models for dynamical systems is of widespread interest in many disciplines. We present a unifying framework for blending mechanistic and machine-learning approaches to identify dynamical systems from noisily and partially observed data. We compare pure data-driven 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 discrete-time 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. parametric-dependence) model error from a learning theory perspective, defining excess risk and generalization error. For ergodic continuous-time systems, we prove that both excess risk and generalization error are bounded above by terms that diminish with the square-root of T T , the time-interval over which training data is specified. Secondly, we study scenarios that benefit from modeling with memory, proving universal approximation theorems for two classes of continuous-time recurrent neural networks (RNNs): both can learn memory-dependent model error, assuming that it is governed by a finite-dimensional hidden variable and that, together, the observed and hidden variables form a continuous-time Markovian system. In addition, we connect one class of RNNs to reservoir computing, thereby relating learning of memory-dependent 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 data-driven and hybrid approaches, finding hybrid methods less datahungry and more parametrically efficient. We also find that, while a continuous-time framing allows for robustness to irregular sampling and desirable domain- interpretability, a discrete-time 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, partially-observed data, and illustrate challenges in representing memory by this approach, and in the training of such models. 

Importance Continuous glucose monitoring (CGM) is associated with improvements in hemoglobin A 1c (HbA 1c ) in youths with type 1 diabetes (T1D); however, youths from minoritized racial and ethnic groups and those with public insurance face greater barriers to CGM access. Early initiation of and access to CGM may reduce disparities in CGM uptake and improve diabetes outcomes. Objective To determine whether HbA 1c decreases differed by ethnicity and insurance status among a cohort of youths newly diagnosed with T1D and provided CGM. Design, Setting, and Participants This cohort study used data from the Teamwork, Targets, Technology, and Tight Control (4T) study, a clinical research program that aims to initiate CGM within 1 month of T1D diagnosis. All youths with new-onset T1D diagnosed between July 25, 2018, and June 15, 2020, at Stanford Children’s Hospital, a single-site, freestanding children’s hospital in California, were approached to enroll in the Pilot-4T study and were followed for 12 months. Data analysis was performed and completed on June 3, 2022. Exposures All eligible participants were offered CGM within 1 month of diabetes diagnosis. Main Outcomes and Measures To assess HbA 1c change over the study period, analyses were stratified by ethnicity (Hispanic vs non-Hispanic) or insurance status (public vs private) to compare the Pilot-4T cohort with a historical cohort of 272 youths diagnosed with T1D between June 1, 2014, and December 28, 2016. Results The Pilot-4T cohort comprised 135 youths, with a median age of 9.7 years (IQR, 6.8-12.7 years) at diagnosis. There were 71 boys (52.6%) and 64 girls (47.4%). Based on self-report, participants’ race was categorized as Asian or Pacific Islander (19 [14.1%]), White (62 [45.9%]), or other race (39 [28.9%]); race was missing or not reported for 15 participants (11.1%). Participants also self-reported their ethnicity as Hispanic (29 [21.5%]) or non-Hispanic (92 [68.1%]). A total of 104 participants (77.0%) had private insurance and 31 (23.0%) had public insurance. Compared with the historical cohort, similar reductions in HbA 1c at 6, 9, and 12 months postdiagnosis were observed for Hispanic individuals (estimated difference, −0.26% [95% CI, −1.05% to 0.43%], −0.60% [−1.46% to 0.21%], and −0.15% [−1.48% to 0.80%]) and non-Hispanic individuals (estimated difference, −0.27% [95% CI, −0.62% to 0.10%], −0.50% [−0.81% to −0.11%], and −0.47% [−0.91% to 0.06%]) in the Pilot-4T cohort. Similar reductions in HbA 1c at 6, 9, and 12 months postdiagnosis were also observed for publicly insured individuals (estimated difference, −0.52% [95% CI, −1.22% to 0.15%], −0.38% [−1.26% to 0.33%], and −0.57% [−2.08% to 0.74%]) and privately insured individuals (estimated difference, −0.34% [95% CI, −0.67% to 0.03%], −0.57% [−0.85% to −0.26%], and −0.43% [−0.85% to 0.01%]) in the Pilot-4T cohort. Hispanic youths in the Pilot-4T cohort had higher HbA 1c at 6, 9, and 12 months postdiagnosis than non-Hispanic youths (estimated difference, 0.28% [95% CI, −0.46% to 0.86%], 0.63% [0.02% to 1.20%], and 1.39% [0.37% to 1.96%]), as did publicly insured youths compared with privately insured youths (estimated difference, 0.39% [95% CI, −0.23% to 0.99%], 0.95% [0.28% to 1.45%], and 1.16% [−0.09% to 2.13%]). Conclusions and Relevance The findings of this cohort study suggest that CGM initiation soon after diagnosis is associated with similar improvements in HbA 1c for Hispanic and non-Hispanic youths as well as for publicly and privately insured youths. These results further suggest that equitable access to CGM soon after T1D diagnosis may be a first step to improve HbA 1c for all youths but is unlikely to eliminate disparities entirely. Trial Registration ClinicalTrials.gov Identifier: NCT04336969