Training Hamiltonian neural networks without backpropagation

Neural networks that synergistically integrate data and physical laws offer great promise in modeling dynamical systems. However, iterative gradient-based optimization of network parameters is often computationally expensive and suffers from slow convergence. In this work, we present a backpropagation-free algorithm to accelerate the training of neural networks for approximating Hamiltonian systems through data-agnostic and data-driven algorithms. We empirically show that data-driven sampling of the network parameters outperforms data-agnostic sampling or the traditional gradient-based iterative optimization of the network parameters when approximating functions with steep gradients or wide input domains. We demonstrate that our approach is more than 100 times faster with CPUs than the traditionally trained Hamiltonian Neural Networks using gradient-based iterative optimization and is more than four orders of magnitude accurate in chaotic examples, including the H\'enon-Heiles system.

Gradient-free training of recurrent neural networks

Recurrent neural networks are a successful neural architecture for many time-dependent problems, including time series analysis, forecasting, and modeling of dynamical systems. Training such networks with backpropagation through time is a notoriously difficult problem because their loss gradients tend to explode or vanish. In this contribution, we introduce a computational approach to construct all weights and biases of a recurrent neural network without using gradient-based methods. The approach is based on a combination of random feature networks and Koopman operator theory for dynamical systems. The hidden parameters of a single recurrent block are sampled at random, while the outer weights are constructed using extended dynamic mode decomposition. This approach alleviates all problems with backpropagation commonly related to recurrent networks. The connection to Koopman operator theory also allows us to start using results in this area to analyze recurrent neural networks. In computational experiments on time series, forecasting for chaotic dynamical systems, and control problems, as well as on weather data, we observe that the training time and forecasting accuracy of the recurrent neural networks we construct are improved when compared to commonly used gradient-based methods.

Accelerating Full Waveform Inversion By Transfer Learning

Full waveform inversion (FWI) is a powerful tool for reconstructing material fields based on sparsely measured data obtained by wave propagation. For specific problems, discretizing the material field with a neural network (NN) improves the robustness and reconstruction quality of the corresponding optimization problem. We call this method NN-based FWI. Starting from an initial guess, the weights of the NN are iteratively updated to fit the simulated wave signals to the sparsely measured data set. For gradient-based optimization, a suitable choice of the initial guess, i.e., a suitable NN weight initialization, is crucial for fast and robust convergence. In this paper, we introduce a novel transfer learning approach to further improve NN-based FWI. This approach leverages supervised pretraining to provide a better NN weight initialization, leading to faster convergence of the subsequent optimization problem. Moreover, the inversions yield physically more meaningful local minima. The network is pretrained to predict the unknown material field using the gradient information from the first iteration of conventional FWI. In our computational experiments on two-dimensional domains, the training data set consists of reference simulations with arbitrarily positioned elliptical voids of different shapes and orientations. We compare the performance of the proposed transfer learning NN-based FWI with three other methods: conventional FWI, NN-based FWI without pretraining and conventional FWI with an initial guess predicted from the pretrained NN. Our results show that transfer learning NN-based FWI outperforms the other methods in terms of convergence speed and reconstruction quality.

On Learning what to Learn: heterogeneous observations of dynamics and establishing (possibly causal) relations among them

Before we attempt to learn a function between two (sets of) observables of a physical process, we must first decide what the inputs and what the outputs of the desired function are going to be. Here we demonstrate two distinct, data-driven ways of initially deciding ``the right quantities'' to relate through such a function, and then proceed to learn it. This is accomplished by processing multiple simultaneous heterogeneous data streams (ensembles of time series) from observations of a physical system: multiple observation processes of the system. We thus determine (a) what subsets of observables are common between the observation processes (and therefore observable from each other, relatable through a function); and (b) what information is unrelated to these common observables, and therefore particular to each observation process, and not contributing to the desired function. Any data-driven function approximation technique can subsequently be used to learn the input-output relation, from k-nearest neighbors and Geometric Harmonics to Gaussian Processes and Neural Networks. Two particular ``twists'' of the approach are discussed. The first has to do with the identifiability of particular quantities of interest from the measurements. We now construct mappings from a single set of observations of one process to entire level sets of measurements of the process, consistent with this single set. The second attempts to relate our framework to a form of causality: if one of the observation processes measures ``now'', while the second observation process measures ``in the future'', the function to be learned among what is common across observation processes constitutes a dynamical model for the system evolution.

Multi-fidelity Gaussian process surrogate modeling for regression problems in physics

One of the main challenges in surrogate modeling is the limited availability of data due to resource constraints associated with computationally expensive simulations. Multi-fidelity methods provide a solution by chaining models in a hierarchy with increasing fidelity, associated with lower error, but increasing cost. In this paper, we compare different multi-fidelity methods employed in constructing Gaussian process surrogates for regression. Non-linear autoregressive methods in the existing literature are primarily confined to two-fidelity models, and we extend these methods to handle more than two levels of fidelity. Additionally, we propose enhancements for an existing method incorporating delay terms by introducing a structured kernel. We demonstrate the performance of these methods across various academic and real-world scenarios. Our findings reveal that multi-fidelity methods generally have a smaller prediction error for the same computational cost as compared to the single-fidelity method, although their effectiveness varies across different scenarios.

Systematic construction of continuous-time neural networks for linear dynamical systems

Discovering a suitable neural network architecture for modeling complex dynamical systems poses a formidable challenge, often involving extensive trial and error and navigation through a high-dimensional hyper-parameter space. In this paper, we discuss a systematic approach to constructing neural architectures for modeling a subclass of dynamical systems, namely, Linear Time-Invariant (LTI) systems. We use a variant of continuous-time neural networks in which the output of each neuron evolves continuously as a solution of a first-order or second-order Ordinary Differential Equation (ODE). Instead of deriving the network architecture and parameters from data, we propose a gradient-free algorithm to compute sparse architecture and network parameters directly from the given LTI system, leveraging its properties. We bring forth a novel neural architecture paradigm featuring horizontal hidden layers and provide insights into why employing conventional neural architectures with vertical hidden layers may not be favorable. We also provide an upper bound on the numerical errors of our neural networks. Finally, we demonstrate the high accuracy of our constructed networks on three numerical examples.

Gappy local conformal auto-encoders for heterogeneous data fusion: in praise of rigidity

Fusing measurements from multiple, heterogeneous, partial sources, observing a common object or process, poses challenges due to the increasing availability of numbers and types of sensors. In this work we propose, implement and validate an end-to-end computational pipeline in the form of a multiple-auto-encoder neural network architecture for this task. The inputs to the pipeline are several sets of partial observations, and the result is a globally consistent latent space, harmonizing (rigidifying, fusing) all measurements. The key enabler is the availability of multiple slightly perturbed measurements of each instance:, local measurement, "bursts", that allows us to estimate the local distortion induced by each instrument. We demonstrate the approach in a sequence of examples, starting with simple two-dimensional data sets and proceeding to a Wi-Fi localization problem and to the solution of a "dynamical puzzle" arising in spatio-temporal observations of the solutions of Partial Differential Equations.

Sampling weights of deep neural networks

We introduce a probability distribution, combined with an efficient sampling algorithm, for weights and biases of fully-connected neural networks. In a supervised learning context, no iterative optimization or gradient computations of internal network parameters are needed to obtain a trained network. The sampling is based on the idea of random feature models. However, instead of a data-agnostic distribution, e.g., a normal distribution, we use both the input and the output training data of the supervised learning problem to sample both shallow and deep networks. We prove that the sampled networks we construct are universal approximators. We also show that our sampling scheme is invariant to rigid body transformations and scaling of the input data. This implies many popular pre-processing techniques are no longer required. For Barron functions, we show that the $L^2$-approximation error of sampled shallow networks decreases with the square root of the number of neurons. In numerical experiments, we demonstrate that sampled networks achieve comparable accuracy as iteratively trained ones, but can be constructed orders of magnitude faster. Our test cases involve a classification benchmark from OpenML, sampling of neural operators to represent maps in function spaces, and transfer learning using well-known architectures.

Data-driven modelling of brain activity using neural networks, Diffusion Maps, and the Koopman operator

We propose a machine-learning approach to model long-term out-of-sample dynamics of brain activity from task-dependent fMRI data. Our approach is a three stage one. First, we exploit Diffusion maps (DMs) to discover a set of variables that parametrize the low-dimensional manifold on which the emergent high-dimensional fMRI time series evolve. Then, we construct reduced-order-models (ROMs) on the embedded manifold via two techniques: Feedforward Neural Networks (FNNs) and the Koopman operator. Finally, for predicting the out-of-sample long-term dynamics of brain activity in the ambient fMRI space, we solve the pre-image problem coupling DMs with Geometric Harmonics (GH) when using FNNs and the Koopman modes per se. For our illustrations, we have assessed the performance of the two proposed schemes using a benchmark fMRI dataset with recordings during a visuo-motor task. The results suggest that just a few (for the particular task, five) non-linear coordinates of the high-dimensional fMRI time series provide a good basis for modelling and out-of-sample prediction of the brain activity. Furthermore, we show that the proposed approaches outperform the one-step ahead predictions of the naive random walk model, which, in contrast to our scheme, relies on the knowledge of the signals in the previous time step. Importantly, we show that the proposed Koopman operator approach provides, for any practical purposes, equivalent results to the FNN-GH approach, thus bypassing the need to train a non-linear map and to use GH to extrapolate predictions in the ambient fMRI space; one can use instead the low-frequency truncation of the DMs function space of L^2-integrable functions, to predict the entire list of coordinate functions in the fMRI space and to solve the pre-image problem.

A Recursively Recurrent Neural Network (R2N2) Architecture for Learning Iterative Algorithms

Meta-learning of numerical algorithms for a given task consist of the data-driven identification and adaptation of an algorithmic structure and the associated hyperparameters. To limit the complexity of the meta-learning problem, neural architectures with a certain inductive bias towards favorable algorithmic structures can, and should, be used. We generalize our previously introduced Runge-Kutta neural network to a recursively recurrent neural network (R2N2) superstructure for the design of customized iterative algorithms. In contrast to off-the-shelf deep learning approaches, it features a distinct division into modules for generation of information and for the subsequent assembly of this information towards a solution. Local information in the form of a subspace is generated by subordinate, inner, iterations of recurrent function evaluations starting at the current outer iterate. The update to the next outer iterate is computed as a linear combination of these evaluations, reducing the residual in this space, and constitutes the output of the network. We demonstrate that regular training of the weight parameters inside the proposed superstructure on input/output data of various computational problem classes yields iterations similar to Krylov solvers for linear equation systems, Newton-Krylov solvers for nonlinear equation systems, and Runge-Kutta integrators for ordinary differential equations. Due to its modularity, the superstructure can be readily extended with functionalities needed to represent more general classes of iterative algorithms traditionally based on Taylor series expansions.