Congratulations to Aditya Saravanakumar and Anantha Narayanan Suresh Babu for successfully clearing the MIT Mechanical Engineering PhD Qualifying Exams. They now begin their journey towards outstanding PhD theses. All the best!

An important challenge in the problem of producing accurate forecasts of multiscale dynamics, including but not limited to weather prediction and ocean modeling, is that these dynamical systems are chaotic in nature. A hallmark of chaotic dynamical systems is that they are highly sensitive to small perturbations in the initial conditions and parameter values. As a result, even the best physics-based computational models, often derived from first principles but limited by varied sources of errors, have limited predictive capabilities for both shorter-term state forecasts and for important longer-term global characteristics of the true system. Observational data, however, provide an avenue to increase predictive capabilities by learning the physics missing from lower-fidelity computational models and reducing their various errors. Recent advances in machine learning, and specifically data-driven knowledge-based prediction, have made this a possibility but even state-of-the-art techniques in this area have not been able to produce short-term forecasts beyond a small multiple of the Lyapunov time of the system, even for simple chaotic systems such as the Lorenz 63 model. In this work, we develop a training framework to apply neural ordinary differential equation-based (nODE) closure models to correct errors in the equations of such dynamical systems. We first identify the key training parameters that have an outsize effect on the learning ability of the neural closure models. We then develop a novel learning algorithm, broadly consisting of adaptive tuning of these parameters, designing dynamic multi-loss objective functions, and an error-targeting batching process. We evaluate and showcase our methodology to the chaotic Balance Equations in an array of increasingly difficult learning settings: first, only the coefficient of one missing term in one perturbed equation; second, one entire missing term in on perturbed equation; third, two missing terms in two perturbed equations; and finally the previous but with a perturbation being two orders of magnitude larger than the state, thereby resulting in a completely different attractor. In each of these cases, our new multi-faceted training approach drastically increases both state-of-the-art state predictability (up to 15 Lyapunov times) and attractor-reproducibility. Finally, we validate our results by comparing them with the predictability limit of the chaotic BE system under different magnitudes of perturbations.

Congratulations to Aman Jalan on his graduation! Aman received an SM from Computational Science and Engineering for his research on “Neural Closure Models for Chaotic Dynamical Systems” with our MSEAS group at MIT.

Improving the predictive capability and computational cost of dynamical models is often at the heart of augmenting computational physics with machine learning (ML). However, most learning results are limited in interpretability and generalization over different computational grid resolutions, initial and boundary conditions, domain geometries, and physical or problem-specific parameters. In the present study, we simultaneously address all these challenges by developing the novel and versatile methodology of unified neural partial delay differential equations. We augment existing/low-fidelity dynamical models directly in their partial differential equation (PDE) forms with both Markovian and non-Markovian neural network (NN) closure parameterizations. The melding of the existing models with NNs in the continuous spatiotemporal space followed by numerical discretization automatically allows for the desired generalizability. The Markovian term is designed to enable extraction of its analytical form and thus provides interpretability. The non-Markovian terms allow accounting for inherently missing time delays needed to represent the real world. Our flexible modeling framework provides full autonomy for the design of the unknown closure terms such as using any linear-, shallow-, or deep-NN architectures, selecting the span of the input function libraries, and using either or both Markovian and non-Markovian closure terms, all in accord with prior knowledge. We obtain adjoint PDEs in the continuous form, thus enabling direct implementation across differentiable and non-differentiable computational physics codes, different ML frameworks, and treatment of nonuniformly-spaced spatiotemporal training data. We demonstrate the new generalized neural closure models (gnCMs) framework using four sets of experiments based on advecting nonlinear waves, shocks, and ocean acidification models. Our learned gnCMs discover missing physics, find leading numerical error terms, discriminate among candidate functional forms in an interpretable fashion, achieve generalization, and compensate for the lack of complexity in simpler models. Finally, we analyze the computational advantages of our new framework.

Mapping the seafloor in the deep ocean is currently performed using sonar systems on surface vessels (low-resolution maps) or undersea vessels (high-resolution maps). Surface-based mapping can cover a much wider search area and is not burdened by the complex logistics required for deploying undersea vessels. However, practical size constraints for a tow body or hull-mounted sonar array result in limits in beamforming and imaging resolution. For cost-effective high-resolution mapping of the deep ocean floor from the surface, a mobile wide-aperture sparse array with subarrays distributed across multiple autonomous surface vessels (ASVs) has been designed. Such a system could enable a surface-based sensor to cover a wide area while achieving high-resolution bathymetry, with resolution cells on the order of 1 m² at a 6 km depth. For coherent 3D imaging, such a system must dynamically track the precise relative position of each boat’s sonar subarray through ocean-induced motions, estimate water column and bottom reflection properties, and mitigate interference from the array sidelobes. Sea testing of this core sparse acoustic array technology has been conducted, and planning is underway for relative navigation testing with ASVs capable of hosting an acoustic subarray.