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Register a new userBeyond Predictions in Neural ODEs: Identification and Interventions
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Added by
Hananeh Aliee
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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Spurred by tremendous success in pattern matching and prediction tasks, researchers increasingly resort to machine learning to aid original scientific discovery. Given large amounts of observational data about a system, can we uncover the rules that govern its evolution? Solving this task holds the great promise of fully understanding the causal interactions and being able to make reliable predictions about the systems behavior under interventions. We take a step towards answering this question for time-series data generated from systems of ordinary differential equations (ODEs). While the governing ODEs might not be identifiable from data alone, we show that combining simple regularization schemes with flexible neural ODEs can robustly recover the dynamics and causal structures from time-series data. Our results on a variety of (non)-linear first and second order systems as well as real data validate our method. We conclude by showing that we can also make accurate predictions under interventions on variables or the system itself.
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Continuous deep learning architectures have recently re-emerged as Neural Ordinary Differential Equations (Neural ODEs). This infinite-depth approach theoretically bridges the gap between deep learning and dynamical systems, offering a novel perspective. However, deciphering the inner working of these models is still an open challenge, as most applications apply them as generic black-box modules. In this work we open the box, further developing the continuous-depth formulation with the aim of clarifying the influence of several design choices on the underlying dynamics.
Modeling a systems temporal behaviour in reaction to external stimuli is a fundamental problem in many areas. Pure Machine Learning (ML) approaches often fail in the small sample regime and cannot provide actionable insights beyond predictions. A promising modification has been to incorporate expert domain knowledge into ML models. The application we consider is predicting the progression of disease under medications, where a plethora of domain knowledge is available from pharmacology. Pharmacological models describe the dynamics of carefully-chosen medically meaningful variables in terms of systems of Ordinary Differential Equations (ODEs). However, these models only describe a limited collection of variables, and these variables are often not observable in clinical environments. To close this gap, we propose the latent hybridisation model (LHM) that integrates a system of expert-designed ODEs with machine-learned Neural ODEs to fully describe the dynamics of the system and to link the expert and latent variables to observable quantities. We evaluated LHM on synthetic data as well as real-world intensive care data of COVID-19 patients. LHM consistently outperforms previous works, especially when few training samples are available such as at the beginning of the pandemic.
Neural Ordinary Differential Equations (ODEs) are elegant reinterpretations of deep networks where continuous time can replace the discrete notion of depth, ODE solvers perform forward propagation, and the adjoint method enables efficient, constant memory backpropagation. Neural ODEs are universal approximators only when they are non-autonomous, that is, the dynamics depends explicitly on time. We propose a novel family of Neural ODEs with time-varying weights, where time-dependence is non-parametric, and the smoothness of weight trajectories can be explicitly controlled to allow a tradeoff between expressiveness and efficiency. Using this enhanced expressiveness, we outperform previous Neural ODE variants in both speed and representational capacity, ultimately outperforming standard ResNet and CNN models on select image classification and video prediction tasks.
This paper proposes the use of spectral element methods citep{canuto_spectral_1988} for fast and accurate training of Neural Ordinary Differential Equations (ODE-Nets; citealp{Chen2018NeuralOD}) for system identification. This is achieved by expressing their dynamics as a truncated series of Legendre polynomials. The series coefficients, as well as the network weights, are computed by minimizing the weighted sum of the loss function and the violation of the ODE-Net dynamics. The problem is solved by coordinate descent that alternately minimizes, with respect to the coefficients and the weights, two unconstrained sub-problems using standard backpropagation and gradient methods. The resulting optimization scheme is fully time-parallel and results in a low memory footprint. Experimental comparison to standard methods, such as backpropagation through explicit solvers and the adjoint technique citep{Chen2018NeuralOD}, on training surrogate models of small and medium-scale dynamical systems shows that it is at least one order of magnitude faster at reaching a comparable value of the loss function. The corresponding testing MSE is one order of magnitude smaller as well, suggesting generalization capabilities increase.
Piecewise linear neural networks can be split into subfunctions, each with its own activation pattern, domain, and empirical error. Empirical error for the full network can be written as an expectation over empirical error of subfunctions. Constructing a generalization bound on subfunction empirical error indicates that the more densely a subfunction is surrounded by training samples in representation space, the more reliable its predictions are. Further, it suggests that models with fewer activation regions generalize better, and models that abstract knowledge to a greater degree generalize better, all else equal. We propose not only a theoretical framework to reason about subfunction error bounds but also a pragmatic way of approximately evaluating it, which we apply to predicting which samples the network will not successfully generalize to. We test our method on detection of misclassification and out-of-distribution samples, finding that it performs competitively in both cases. In short, some network activation patterns are associated with higher reliability than others, and these can be identified using subfunction error bounds.
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