Ashutosh Singh

Ashutosh Singh, Ashish Singh, Tales Imbiriba et al.

Approaches based on Koopman operators have shown great promise in forecasting time series data generated by complex nonlinear dynamical systems (NLDS). Although such approaches are able to capture the latent state representation of a NLDS, they still face difficulty in long term forecasting when applied to real world data. Specifically many real-world NLDS exhibit time-varying behavior, leading to nonstationarity that is hard to capture with such models. Furthermore they lack a systematic data-driven approach to perform data assimilation, that is, exploiting noisy measurements on the fly in the forecasting task. To alleviate the above issues, we propose a Koopman operator-based approach (named KODA - Koopman Operator with Data Assimilation) that integrates forecasting and data assimilation in NLDS. In particular we use a Fourier domain filter to disentangle the data into a physical component whose dynamics can be accurately represented by a Koopman operator, and residual dynamics that represents the local or time varying behavior that are captured by a flexible and learnable recursive model. We carefully design an architecture and training criterion that ensures this decomposition lead to stable and long-term forecasts. Moreover, we introduce a course correction strategy to perform data assimilation with new measurements at inference time. The proposed approach is completely data-driven and can be learned end-to-end. Through extensive experimental comparisons we show that KODA outperforms existing state of the art methods on multiple time series benchmarks such as electricity, temperature, weather, lorenz 63 and duffing oscillator demonstrating its superior performance and efficacy along the three tasks a) forecasting, b) data assimilation and c) state prediction.

Ashutosh Singh, Ricardo Augusto Borsoi, Deniz Erdogmus et al.

Recent advances in the theory of Neural Operators (NOs) have enabled fast and accurate computation of the solutions to complex systems described by partial differential equations (PDEs). Despite their great success, current NO-based solutions face important challenges when dealing with spatio-temporal PDEs over long time scales. Specifically, the current theory of NOs does not present a systematic framework to perform data assimilation and efficiently correct the evolution of PDE solutions over time based on sparsely sampled noisy measurements. In this paper, we propose a learning-based state-space approach to compute the solution operators to infinite-dimensional semilinear PDEs. Exploiting the structure of semilinear PDEs and the theory of nonlinear observers in function spaces, we develop a flexible recursive method that allows for both prediction and data assimilation by combining prediction and correction operations. The proposed framework is capable of producing fast and accurate predictions over long time horizons, dealing with irregularly sampled noisy measurements to correct the solution, and benefits from the decoupling between the spatial and temporal dynamics of this class of PDEs. We show through experiments on the Kuramoto-Sivashinsky, Navier-Stokes and Korteweg-de Vries equations that the proposed model is robust to noise and can leverage arbitrary amounts of measurements to correct its prediction over a long time horizon with little computational overhead.

Christiana Westlin, Ashutosh Singh, Deniz Erdogmus et al.

In the science of emotion, it is widely assumed that folk emotion categories form a biological and psychological typology, and studies are routinely designed and analyzed to identify emotion-specific patterns. This approach shapes the observations that studies report, ultimately reinforcing the assumption that guided the investigation. Here, we reanalyzed data from one such typologically-guided study that reported mappings between individual brain patterns and group-averaged ratings of 34 emotion categories. Our reanalysis was guided by an alternative view of emotion categories as populations of variable, situated instances, and which predicts a priori that there will be significant variation in brain patterns within a category across instances. Correspondingly, our analysis made minimal assumptions about the structure of the variance present in the data. As predicted, we did not observe the original mappings and instead observed significant variation across individuals. These findings demonstrate how starting assumptions can ultimately impact scientific conclusions and suggest that a hypothesis must be supported using multiple analytic methods before it is taken seriously.