Mayuresh V. Kothare’s research while affiliated with Lehigh University and other places

Multiple-shooting is a parameter estimation approach for ordinary differential equations. In this approach, the trajectory is broken into small intervals, each of which can be integrated independently. Equality constraints are then applied to eliminate the shooting gap between the end of the previous trajectory and the start of the next trajectory. Unlike single-shooting, multiple-shooting is more stable, especially for highly oscillatory and long trajectories. In the context of neural ordinary differential equations, multiple-shooting is not widely used due to the challenge of incorporating general equality constraints. In this work, we propose a condensing-based approach to incorporate these shooting equality constraints while training a multiple-shooting neural ordinary differential equation (MS-NODE) using first-order optimization methods such as Adam.

Inverse problem or parameter estimation of ordinary differential equations is a process of obtaining the best parameters using experimental measurements of the states. Single (Multiple)-shooting is a type of sequential optimization method that minimizes the error in the measured and numerically integrated states. However, this requires computing sensitivities i.e. the derivatives of states with respect to the parameters over the numerical integrator, which can get computationally expensive. To address this challenge, many interpolation-based approaches have been proposed to either reduce the computational cost of sensitivity calculations or eliminate their need. In this paper, we use a bi-level optimization framework that leverages interpolation and exploits the structure of the differential equation to solve an inner convex optimization problem. We apply this method to two different problem formulations. First, parameter estimation for differential equations, and delayed differential equations, where the model structure is known but the parameters are unknown. Second, model discovery problems, where both the model structure and parameters are unknown.

Developing data-driven kinetic models from reaction data is valuable for inferring the underlying reactions and designing reactive processes without needing first-principles models. However, recently developed techniques to learn interpretable dynamical models from data are susceptible to inherent experimental noise, especially in reaction kinetics data. Here, we address these issues by (1) employing a new derivative-free technique for sparse identification of dynamical equations that approximates the integral rather than the derivative (which we call as DF-SINDy) and (2) including domain information such as mass balance and chemistry information. We demonstrate this using retrospective examples to recover the true (known) governing equations from synthetic data under varying noise levels, sampling frequencies, and number of experiments. We observe that (1) models discovered from DF-SINDy have lower errors than those discovered from SINDy (Proc. Natl. Acad. Sci. U.S.A.2016, 113, 3932−3937, DOI: 10.1073/pnas.151738411327035946 ) and (2) adding domain knowledge further helps recover correct terms, thereby improving the reliability of the interpretations obtained from these models. This work is chemistry agnostic and represents a step toward developing domain-informed interpretable kinetic models for complex reaction networks.

Differential Dynamic Programming (DDP) is one of the indirect methods for solving an optimal control problem. Several extensions to DDP have been proposed to add stagewise state and control constraints, which can mainly be classified as augmented lagrangian methods, active set methods, and barrier methods. In this paper, we use an interior point method, which is a type of barrier method, to incorporate arbitrary stagewise equality and inequality state and control constraints. We also provide explicit update formulas for all the involved variables. Finally, we apply this algorithm to example systems such as the inverted pendulum, a continuously stirred tank reactor, car parking, and obstacle avoidance.

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Objective. Vagus nerve stimulation (VNS) is being investigated as a potential therapy for cardiovascular diseases including heart failure, cardiac arrhythmia, and hypertension. The lack of a systematic approach for controlling and tuning the VNS parameters poses a significant challenge. Closed-loop VNS strategies combined with artificial intelligence (AI) approaches offer a framework for systematically learning and adapting the optimal stimulation parameters. In this study, we presented an interactive AI framework using reinforcement learning (RL) for automated data-driven design of closed-loop VNS control systems in a computational study. Approach. Multiple simulation environments with a standard application programming interface were developed to facilitate the design and evaluation of the automated data-driven closed-loop VNS control systems. These environments simulate the hemodynamic response to multi-location VNS using biophysics-based computational models of healthy and hypertensive rat cardiovascular systems in resting and exercise states. We designed and implemented the RL-based closed-loop VNS control frameworks in the context of controlling the heart rate and the mean arterial pressure for a set point tracking task. Our experimental design included two approaches; a general policy using deep RL algorithms and a sample-efficient adaptive policy using probabilistic inference for learning and control. Main results. Our simulation results demonstrated the capabilities of the closed-loop RL-based approaches to learn optimal VNS control policies and to adapt to variations in the target set points and the underlying dynamics of the cardiovascular system. Our findings highlighted the trade-off between sample-efficiency and generalizability, providing insights for proper algorithm selection. Finally, we demonstrated that transfer learning improves the sample efficiency of deep RL algorithms allowing the development of more efficient and personalized closed-loop VNS systems. Significance. We demonstrated the capability of RL-based closed-loop VNS systems. Our approach provided a systematic adaptable framework for learning control strategies without requiring prior knowledge about the underlying dynamics.

Vagal nerve stimulation (VNS) is currently under investigation for the treatment of various cardiovascular diseases including heart failure, arrhythmia, and hypertension. In preclinical and clinical studies, VNS stimulation parameters are heuristically determined in the open loop. However, its therapeutic efficacy remains inconclusive, strongly suggesting the need for a closed-loop approach to optimize patient-specific stimulation parameters. In this paper, we develop a multiple model predictive control (MMPC) algorithm for automated regulation of heart rate (HR) and mean arterial pressure by optimally adjusting the amplitude and frequency of electrical pulses applied to three locations of the vagal nerve. The multiple local models are identified from our previously reported pulsatile rat cardiac model that emulates symptoms of hypertension in rest and exercise states. The computational expense of the proposed method is verified in simulation with rigorous hardware-in-the-loop (HIL) implementation.