Hybrid variable spiking graph neural networks for energy-efficient scientific machine learning

Graph-based representations for samples of computational mechanics-related datasets can prove instrumental when dealing with problems like irregular domains or molecular structures of materials, etc. To effectively analyze and process such datasets, deep learning offers Graph Neural Networks (GNNs) that utilize techniques like message-passing within their architecture. The issue, however, is that as the individual graph scales and/ or GNN architecture becomes increasingly complex, the increased energy budget of the overall deep learning model makes it unsustainable and restricts its applications in applications like edge computing. To overcome this, we propose in this paper Hybrid Variable Spiking Graph Neural Networks (HVS-GNNs) that utilize Variable Spiking Neurons (VSNs) within their architecture to promote sparse communication and hence reduce the overall energy budget. VSNs, while promoting sparse event-driven computations, also perform well for regression tasks, which are often encountered in computational mechanics applications and are the main target of this paper. Three examples dealing with prediction of mechanical properties of material based on microscale/ mesoscale structures are shown to test the performance of the proposed HVS-GNNs in regression tasks. We have also compared the performance of HVS-GNN architectures with the performance of vanilla GNNs and GNNs utilizing leaky integrate and fire neurons. The results produced show that HVS-GNNs perform well for regression tasks, all while promoting sparse communication and, hence, energy efficiency.

G$^{2}$TR: Generalized Grounded Temporal Reasoning for Robot Instruction Following by Combining Large Pre-trained Models

Consider the scenario where a human cleans a table and a robot observing the scene is instructed with the task "Remove the cloth using which I wiped the table". Instruction following with temporal reasoning requires the robot to identify the relevant past object interaction, ground the object of interest in the present scene, and execute the task according to the human's instruction. Directly grounding utterances referencing past interactions to grounded objects is challenging due to the multi-hop nature of references to past interactions and large space of object groundings in a video stream observing the robot's workspace. Our key insight is to factor the temporal reasoning task as (i) estimating the video interval associated with event reference, (ii) performing spatial reasoning over the interaction frames to infer the intended object (iii) semantically track the object's location till the current scene to enable future robot interactions. Our approach leverages existing large pre-trained models (which possess inherent generalization capabilities) and combines them appropriately for temporal grounding tasks. Evaluation on a video-language corpus acquired with a robot manipulator displaying rich temporal interactions in spatially-complex scenes displays an average accuracy of 70.10%. The dataset, code, and videos are available at https://reail-iitdelhi.github.io/temporalreasoning.github.io/ .

Towards Gaussian Process for operator learning: an uncertainty aware resolution independent operator learning algorithm for computational mechanics

The growing demand for accurate, efficient, and scalable solutions in computational mechanics highlights the need for advanced operator learning algorithms that can efficiently handle large datasets while providing reliable uncertainty quantification. This paper introduces a novel Gaussian Process (GP) based neural operator for solving parametric differential equations. The approach proposed leverages the expressive capability of deterministic neural operators and the uncertainty awareness of conventional GP. In particular, we propose a ``neural operator-embedded kernel'' wherein the GP kernel is formulated in the latent space learned using a neural operator. Further, we exploit a stochastic dual descent (SDD) algorithm for simultaneously training the neural operator parameters and the GP hyperparameters. Our approach addresses the (a) resolution dependence and (b) cubic complexity of traditional GP models, allowing for input-resolution independence and scalability in high-dimensional and non-linear parametric systems, such as those encountered in computational mechanics. We apply our method to a range of non-linear parametric partial differential equations (PDEs) and demonstrate its superiority in both computational efficiency and accuracy compared to standard GP models and wavelet neural operators. Our experimental results highlight the efficacy of this framework in solving complex PDEs while maintaining robustness in uncertainty estimation, positioning it as a scalable and reliable operator-learning algorithm for computational mechanics.

Harnessing physics-informed operators for high-dimensional reliability analysis problems

Reliability analysis is a formidable task, particularly in systems with a large number of stochastic parameters. Conventional methods for quantifying reliability often rely on extensive simulations or experimental data, which can be costly and time-consuming, especially when dealing with systems governed by complex physical laws which necessitates computationally intensive numerical methods such as finite element or finite volume techniques. On the other hand, surrogate-based methods offer an efficient alternative for computing reliability by approximating the underlying model from limited data. Neural operators have recently emerged as effective surrogates for modelling physical systems governed by partial differential equations. These operators can learn solutions to PDEs for varying inputs and parameters. Here, we investigate the efficacy of the recently developed physics-informed wavelet neural operator in solving reliability analysis problems. In particular, we investigate the possibility of using physics-informed operator for solving high-dimensional reliability analysis problems, while bypassing the need for any simulation. Through four numerical examples, we illustrate that physics-informed operator can seamlessly solve high-dimensional reliability analysis problems with reasonable accuracy, while eliminating the need for running expensive simulations.

Spatio-spectral graph neural operator for solving computational mechanics problems on irregular domain and unstructured grid

Scientific machine learning has seen significant progress with the emergence of operator learning. However, existing methods encounter difficulties when applied to problems on unstructured grids and irregular domains. Spatial graph neural networks utilize local convolution in a neighborhood to potentially address these challenges, yet they often suffer from issues such as over-smoothing and over-squashing in deep architectures. Conversely, spectral graph neural networks leverage global convolution to capture extensive features and long-range dependencies in domain graphs, albeit at a high computational cost due to Eigenvalue decomposition. In this paper, we introduce a novel approach, referred to as Spatio-Spectral Graph Neural Operator (Sp$^2$GNO) that integrates spatial and spectral GNNs effectively. This framework mitigates the limitations of individual methods and enables the learning of solution operators across arbitrary geometries, thus catering to a wide range of real-world problems. Sp$^2$GNO demonstrates exceptional performance in solving both time-dependent and time-independent partial differential equations on regular and irregular domains. Our approach is validated through comprehensive benchmarks and practical applications drawn from computational mechanics and scientific computing literature.

Discovering governing equation in structural dynamics from acceleration-only measurements

Over the past few years, equation discovery has gained popularity in different fields of science and engineering. However, existing equation discovery algorithms rely on the availability of noisy measurements of the state variables (i.e., displacement {and velocity}). This is a major bottleneck in structural dynamics, where we often only have access to acceleration measurements. To that end, this paper introduces a novel equation discovery algorithm for discovering governing equations of dynamical systems from acceleration-only measurements. The proposed algorithm employs a library-based approach for equation discovery. To enable equation discovery from acceleration-only measurements, we propose a novel Approximate Bayesian Computation (ABC) model that prioritizes parsimonious models. The efficacy of the proposed algorithm is illustrated using {four} structural dynamics examples that include both linear and nonlinear dynamical systems. The case studies presented illustrate the possible application of the proposed approach for equation discovery of dynamical systems from acceleration-only measurements.

Generative flow induced neural architecture search: Towards discovering optimal architecture in wavelet neural operator

We propose a generative flow-induced neural architecture search algorithm. The proposed approach devices simple feed-forward neural networks to learn stochastic policies to generate sequences of architecture hyperparameters such that the generated states are in proportion with the reward from the terminal state. We demonstrate the efficacy of the proposed search algorithm on the wavelet neural operator (WNO), where we learn a policy to generate a sequence of hyperparameters like wavelet basis and activation operators for wavelet integral blocks. While the trajectory of the generated wavelet basis and activation sequence is cast as flow, the policy is learned by minimizing the flow violation between each state in the trajectory and maximizing the reward from the terminal state. In the terminal state, we train WNO simultaneously to guide the search. We propose to use the exponent of the negative of the WNO loss on the validation dataset as the reward function. While the grid search-based neural architecture generation algorithms foresee every combination, the proposed framework generates the most probable sequence based on the positive reward from the terminal state, thereby reducing exploration time. Compared to reinforcement learning schemes, where complete episodic training is required to get the reward, the proposed algorithm generates the hyperparameter trajectory sequentially. Through four fluid mechanics-oriented problems, we illustrate that the learned policies can sample the best-performing architecture of the neural operator, thereby improving the performance of the vanilla wavelet neural operator.

Neural Operator induced Gaussian Process framework for probabilistic solution of parametric partial differential equations

The study of neural operators has paved the way for the development of efficient approaches for solving partial differential equations (PDEs) compared with traditional methods. However, most of the existing neural operators lack the capability to provide uncertainty measures for their predictions, a crucial aspect, especially in data-driven scenarios with limited available data. In this work, we propose a novel Neural Operator-induced Gaussian Process (NOGaP), which exploits the probabilistic characteristics of Gaussian Processes (GPs) while leveraging the learning prowess of operator learning. The proposed framework leads to improved prediction accuracy and offers a quantifiable measure of uncertainty. The proposed framework is extensively evaluated through experiments on various PDE examples, including Burger's equation, Darcy flow, non-homogeneous Poisson, and wave-advection equations. Furthermore, a comparative study with state-of-the-art operator learning algorithms is presented to highlight the advantages of NOGaP. The results demonstrate superior accuracy and expected uncertainty characteristics, suggesting the promising potential of the proposed framework.

MD-NOMAD: Mixture density nonlinear manifold decoder for emulating stochastic differential equations and uncertainty propagation

We propose a neural operator framework, termed mixture density nonlinear manifold decoder (MD-NOMAD), for stochastic simulators. Our approach leverages an amalgamation of the pointwise operator learning neural architecture nonlinear manifold decoder (NOMAD) with mixture density-based methods to estimate conditional probability distributions for stochastic output functions. MD-NOMAD harnesses the ability of probabilistic mixture models to estimate complex probability and the high-dimensional scalability of pointwise neural operator NOMAD. We conduct empirical assessments on a wide array of stochastic ordinary and partial differential equations and present the corresponding results, which highlight the performance of the proposed framework.

PhyPlan: Compositional and Adaptive Physical Task Reasoning with Physics-Informed Skill Networks for Robot Manipulators

Given the task of positioning a ball-like object to a goal region beyond direct reach, humans can often throw, slide, or rebound objects against the wall to attain the goal. However, enabling robots to reason similarly is non-trivial. Existing methods for physical reasoning are data-hungry and struggle with complexity and uncertainty inherent in the real world. This paper presents PhyPlan, a novel physics-informed planning framework that combines physics-informed neural networks (PINNs) with modified Monte Carlo Tree Search (MCTS) to enable embodied agents to perform dynamic physical tasks. PhyPlan leverages PINNs to simulate and predict outcomes of actions in a fast and accurate manner and uses MCTS for planning. It dynamically determines whether to consult a PINN-based simulator (coarse but fast) or engage directly with the actual environment (fine but slow) to determine optimal policy. Evaluation with robots in simulated 3D environments demonstrates the ability of our approach to solve 3D-physical reasoning tasks involving the composition of dynamic skills. Quantitatively, PhyPlan excels in several aspects: (i) it achieves lower regret when learning novel tasks compared to state-of-the-art, (ii) it expedites skill learning and enhances the speed of physical reasoning, (iii) it demonstrates higher data efficiency compared to a physics un-informed approach.