Srinivas Rangarajan

Siddharth Prabhu, Srinivas Rangarajan, Mayuresh Kothare

Inverse problem or parameter estimation of ordinary differential equations (ODEs), the iterative process of minimizing the mismatch between model-predicted and experimental states by tuning the parameter values within an optimization formulation, is commonplace in chemical engineering applications. A popular method for parameter estimation is sequential optimization (single-shooting), which numerically integrates the ODE in each iteration. However, computing the gradients for the optimization steps requires calculating sensitivities, i.e., the derivatives of states with respect to the parameters, through the numerical integrator, which can be computationally expensive. In this work, we use interpolation to reduce the cost of these sensitivity calculations. Leveraging this interpolation, we also propose a bi-level optimization framework that exploits the structure of the differential equations and solves a convex inner problem. We apply this framework to examples spanning conventional parameter estimation and the emerging concept of data-driven dynamic model discovery. We show that our approach not only estimates the correct parameters for benchmark problems, but can also be readily extended to delay, stiff, and partially observed differential equations without major modifications.

Siddharth Prabhu, Srinivas Rangarajan, Mayuresh Kothare

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.

Chao Chen, Yifan Shen, Guixiang Ma et al.

Learning to compare two objects are essential in applications, such as digital forensics, face recognition, and brain network analysis, especially when labeled data is scarce and imbalanced. As these applications make high-stake decisions and involve societal values like fairness and transparency, it is critical to explain the learned models. We aim to study post-hoc explanations of Siamese networks (SN) widely used in learning to compare. We characterize the instability of gradient-based explanations due to the additional compared object in SN, in contrast to architectures with a single input instance. We propose an optimization framework that derives global invariance from unlabeled data using self-learning to promote the stability of local explanations tailored for specific query-reference pairs. The optimization problems can be solved using gradient descent-ascent (GDA) for constrained optimization, or SGD for KL-divergence regularized unconstrained optimization, with convergence proofs, especially when the objective functions are nonconvex due to the Siamese architecture. Quantitative results and case studies on tabular and graph data from neuroscience and chemical engineering show that the framework respects the self-learned invariance while robustly optimizing the faithfulness and simplicity of the explanation. We further demonstrate the convergence of GDA experimentally.