We study aleatoric and epistemic uncertainty estimation in a learned regressive system dynamics model. Disentangling aleatoric uncertainty (the inherent randomness of the system) from epistemic uncertainty (the lack of data) is crucial for downstream tasks such as risk-aware control and reinforcement learning, efficient exploration, and robust policy transfer. While existing approaches like Gaussian Processes, Bayesian networks, and model ensembles are widely adopted, they suffer from either high computational complexity or inaccurate uncertainty estimation. To address these limitations, we propose the Compressed Data Representation Model (CDRM), a framework that learns a neural network encoding of the data distribution and enables direct sampling from the output distribution. Our approach incorporates a novel inference procedure based on Langevin dynamics sampling, allowing CDRM to predict arbitrary output distributions rather than being constrained to a Gaussian prior. Theoretical analysis provides the conditions where CDRM achieves better memory and computational complexity compared to bin-based compression methods. Empirical evaluations show that CDRM demonstrates a superior capability to identify aleatoric and epistemic uncertainties separately, achieving AUROCs of 0.8876 and 0.9981 on a single test set containing a mixture of both uncertainties. Qualitative results further show that CDRM's capability extends to datasets with multimodal output distributions, a challenging scenario where existing methods consistently fail. Code and supplementary materials are available at https://github.com/ryeii/CDRM.
We introduce a simultaneous and meshfree topology optimization (TO) framework based on physics-informed Gaussian processes (GPs). Our framework endows all design and state variables via GP priors which have a shared, multi-output mean function that is parametrized via a customized deep neural network (DNN). The parameters of this mean function are estimated by minimizing a multi-component loss function that depends on the performance metric, design constraints, and the residuals on the state equations. Our TO approach yields well-defined material interfaces and has a built-in continuation nature that promotes global optimality. Other unique features of our approach include (1) its customized DNN which, unlike fully connected feed-forward DNNs, has a localized learning capacity that enables capturing intricate topologies and reducing residuals in high gradient fields, (2) its loss function that leverages localized weights to promote solution accuracy around interfaces, and (3) its use of curriculum training to avoid local optimality.To demonstrate the power of our framework, we validate it against commercial TO package COMSOL on three problems involving dissipated power minimization in Stokes flow.
Knowledge of the force time history of a structure is essential to assess its behaviour, ensure safety and maintain reliability. However, direct measurement of external forces is often challenging due to sensor limitations, unknown force characteristics, or inaccessible load points. This paper presents an efficient dynamic load reconstruction method using physics-informed Gaussian processes (GP) based on frequency-sparse Fourier basis functions. The GP's covariance matrices are built using the description of the system dynamics, and the model is trained using structural response measurements. This provides support and interpretability to the machine learning model, in contrast to purely data-driven methods. In addition, the model filters out irrelevant components in the Fourier basis function by leveraging the sparsity of structural responses in the frequency domain, thereby reducing computational complexity during optimization. The trained model for structural responses is then integrated with the differential equation for a harmonic oscillator, creating a probabilistic dynamic load model that predicts load patterns without requiring force data during training. The model's effectiveness is validated through two case studies: a numerical model of a wind-excited 76-story building and an experiment using a physical scale model of the Lilleb{\ae}lt Bridge in Denmark, excited by a servo motor. For both cases, validation of the reconstructed forces is provided using comparison metrics for several signal properties. The developed model holds potential for applications in structural health monitoring, damage prognosis, and load model validation.
We perform Bayesian optimization using a Gaussian process perspective on Bayesian Additive Regression Trees (BART). Our BART Kernel (BARK) uses tree agreement to define a posterior over piecewise-constant functions, and we explore the space of tree kernels using a Markov chain Monte Carlo approach. Where BART only samples functions, the resulting BARK model obtains samples of Gaussian processes defining distributions over functions, which allow us to build acquisition functions for Bayesian optimization. Our tree-based approach enables global optimization over the surrogate, even for mixed-feature spaces. Moreover, where many previous tree-based kernels provide uncertainty quantification over function values, our sampling scheme captures uncertainty over the tree structure itself. Our experiments show the strong performance of BARK on both synthetic and applied benchmarks, due to the combination of our fully Bayesian surrogate and the optimization procedure.
Terrain analysis is critical for the practical application of ground mobile robots in real-world tasks, especially in outdoor unstructured environments. In this paper, we propose a novel spatial-temporal traversability assessment method, which aims to enable autonomous robots to effectively navigate through complex terrains. Our approach utilizes sparse Gaussian processes (SGP) to extract geometric features (curvature, gradient, elevation, etc.) directly from point cloud scans. These features are then used to construct a high-resolution local traversability map. Then, we design a spatial-temporal Bayesian Gaussian kernel (BGK) inference method to dynamically evaluate traversability scores, integrating historical and real-time data while considering factors such as slope, flatness, gradient, and uncertainty metrics. GPU acceleration is applied in the feature extraction step, and the system achieves real-time performance. Extensive simulation experiments across diverse terrain scenarios demonstrate that our method outperforms SOTA approaches in both accuracy and computational efficiency. Additionally, we develop an autonomous navigation framework integrated with the traversability map and validate it with a differential driven vehicle in complex outdoor environments. Our code will be open-source for further research and development by the community, https://github.com/ZJU-FAST-Lab/FSGP_BGK.
In green security, defenders must forecast adversarial behavior, such as poaching, illegal logging, and illegal fishing, to plan effective patrols. These behavior are often highly uncertain and complex. Prior work has leveraged game theory to design robust patrol strategies to handle uncertainty, but existing adversarial behavior models primarily rely on Gaussian processes or linear models, which lack the expressiveness needed to capture intricate behavioral patterns. To address this limitation, we propose a conditional diffusion model for adversary behavior modeling, leveraging its strong distribution-fitting capabilities. To the best of our knowledge, this is the first application of diffusion models in the green security domain. Integrating diffusion models into game-theoretic optimization, however, presents new challenges, including a constrained mixed strategy space and the need to sample from an unnormalized distribution to estimate utilities. To tackle these challenges, we introduce a mixed strategy of mixed strategies and employ a twisted Sequential Monte Carlo (SMC) sampler for accurate sampling. Theoretically, our algorithm is guaranteed to converge to an epsilon equilibrium with high probability using a finite number of iterations and samples. Empirically, we evaluate our approach on both synthetic and real-world poaching datasets, demonstrating its effectiveness.
Bayesian Optimization (BO) with Gaussian Processes relies on optimizing an acquisition function to determine sampling. We investigate the advantages and disadvantages of using a deterministic global solver (MAiNGO) compared to conventional local and stochastic global solvers (L-BFGS-B and multi-start, respectively) for the optimization of the acquisition function. For CPU efficiency, we set a time limit for MAiNGO, taking the best point as optimal. We perform repeated numerical experiments, initially using the Muller-Brown potential as a benchmark function, utilizing the lower confidence bound acquisition function; we further validate our findings with three alternative benchmark functions. Statistical analysis reveals that when the acquisition function is more exploitative (as opposed to exploratory), BO with MAiNGO converges in fewer iterations than with the local solvers. However, when the dataset lacks diversity, or when the acquisition function is overly exploitative, BO with MAiNGO, compared to the local solvers, is more likely to converge to a local rather than a global ly near-optimal solution of the black-box function. L-BFGS-B and multi-start mitigate this risk in BO by introducing stochasticity in the selection of the next sampling point, which enhances the exploration of uncharted regions in the search space and reduces dependence on acquisition function hyperparameters. Ultimately, suboptimal optimization of poorly chosen acquisition functions may be preferable to their optimal solution. When the acquisition function is more exploratory, BO with MAiNGO, multi-start, and L-BFGS-B achieve comparable probabilities of convergence to a globally near-optimal solution (although BO with MAiNGO may require more iterations to converge under these conditions).
With the digitalization of power grids, physical equations become insufficient to describe the network's behavior, and realistic but time-consuming simulators must be used. Numerical experiments, such as safety validation, that involve simulating a large number of scenarios become computationally intractable. A popular solution to reduce the computational burden is to learn a surrogate model of the simulator with Machine Learning (ML) and then conduct the experiment directly on the fast-to-evaluate surrogate model. Among the various ML possibilities for building surrogate models, Gaussian processes (GPs) emerged as a popular solution due to their flexibility, data efficiency, and interpretability. Their probabilistic nature enables them to provide both predictions and uncertainty quantification (UQ). This paper starts with a discussion on the interest of using GPs to approximate power grid simulators and fasten numerical experiments. Such simulators, however, often violate the GP's underlying Gaussian assumption, leading to poor approximations. To address this limitation, an approach that consists in adding an adaptive residual uncertainty term to the UQ is proposed. It enables the GP to remain accurate and reliable despite the simulator's non-Gaussian behaviors. This approach is successfully applied to the certification of the proper functioning of a congestion management controller, with over 98% of simulations avoided.
Physics-informed machine learning is one of the most commonly used methods for fusing physical knowledge in the form of partial differential equations with experimental data. The idea is to construct a loss function where the physical laws take the place of a regularizer and minimize it to reconstruct the underlying physical fields and any missing parameters. However, there is a noticeable lack of a direct connection between physics-informed loss functions and an overarching Bayesian framework. In this work, we demonstrate that Brownian bridge Gaussian processes can be viewed as a softly-enforced physics-constrained prior for the Poisson equation. We first show equivalence between the variational form of the physics-informed loss function for the Poisson equation and a kernel ridge regression objective. Then, through the connection between Gaussian process regression and kernel methods, we identify a Gaussian process for which the posterior mean function and physics-informed loss function minimizer agree. This connection allows us to probe different theoretical questions, such as convergence and behavior of inverse problems. We also connect the method to the important problem of identifying model-form error in applications.
Advanced slot and winding designs are imperative to create future high performance electrical machines (EM). As a result, the development of methods to design and improve slot filling factor (SFF) has attracted considerable research. Recent developments in manufacturing processes, such as additive manufacturing and alternative materials, has also highlighted a need for novel high-fidelity design techniques to develop high performance complex geometries and topologies. This study therefore introduces a novel physics-informed machine learning (PIML) design optimization process for improving SFF in traction electrical machines used in electric vehicles. A maximum entropy sampling algorithm (MESA) is used to seed a physics-informed Bayesian optimization (PIBO) algorithm, where the target function and its approximations are produced by Gaussian processes (GP)s. The proposed PIBO-MESA is coupled with a 2D finite element model (FEM) to perform a GP-based surrogate and provide the first demonstration of the optimal combination of complex design variables for an electrical machine. Significant computational gains were achieved using the new PIBO-MESA approach, which is 45% faster than existing stochastic methods, such as the non-dominated sorting genetic algorithm II (NSGA-II). The FEM results confirm that the new design optimization process and keystone shaped wires lead to a higher SFF (i.e. by 20%) and electromagnetic improvements (e.g. maximum torque by 12%) with similar resistivity. The newly developed PIBO-MESA design optimization process therefore presents significant benefits in the design of high-performance electric machines, with reduced development time and costs.
We adopt Gaussian Processes (GPs) as latent functions for probabilistic forecasting of intermittent time series. The model is trained in a Bayesian framework that accounts for the uncertainty about the latent function and marginalizes it out when making predictions. We couple the latent GP variable with two types of forecast distributions: the negative binomial (NegBinGP) and the Tweedie distribution (TweedieGP). While the negative binomial has already been used in forecasting intermittent time series, this is the first time in which a fully parameterized Tweedie density is used for intermittent time series. We properly evaluate the Tweedie density, which has both a point mass at zero and heavy tails, avoiding simplifying assumptions made in existing models. We test our models on thousands of intermittent count time series. Results show that our models provide consistently better probabilistic forecasts than the competitors. In particular, TweedieGP obtains the best estimates of the highest quantiles, thus showing that it is more flexible than NegBinGP.
This work proposes a scalable probabilistic latent variable model based on Gaussian processes (Lawrence, 2004) in the context of multiple observation spaces. We focus on an application in astrophysics where data sets typically contain both observed spectral features and scientific properties of astrophysical objects such as galaxies or exoplanets. In our application, we study the spectra of very luminous galaxies known as quasars, along with their properties, such as the mass of their central supermassive black hole, accretion rate, and luminosity-resulting in multiple observation spaces. A single data point is then characterized by different classes of observations, each with different likelihoods. Our proposed model extends the baseline stochastic variational Gaussian process latent variable model (GPLVM) introduced by Lalchand et al. (2022) to this setting, proposing a seamless generative model where the quasar spectra and scientific labels can be generated simultaneously using a shared latent space as input to different sets of Gaussian process decoders, one for each observation space. Additionally, this framework enables training in a missing data setting where a large number of dimensions per data point may be unknown or unobserved. We demonstrate high-fidelity reconstructions of the spectra and scientific labels during test-time inference and briefly discuss the scientific interpretations of the results, along with the significance of such a generative model.
Many real-world applications rely on land surface temperature (LST) data at high spatiotemporal resolution. In complex urban areas, LST exhibits significant variations, fluctuating dramatically within and across city blocks. Landsat provides high spatial resolution data at 100 meters but is limited by long revisit time, with cloud cover further disrupting data collection. Here, we propose DELAG, a deep ensemble learning method that integrates annual temperature cycles and Gaussian processes, to reconstruct Landsat LST in complex urban areas. Leveraging the cross-track characteristics and dual-satellite operation of Landsat since 2021, we further enhance data availability to 4 scenes every 16 days. We select New York City, London and Hong Kong from three different continents as study areas. Experiments show that DELAG successfully reconstructed LST in the three cities under clear-sky (RMSE = 0.73-0.96 K) and heavily-cloudy (RMSE = 0.84-1.62 K) situations, superior to existing methods. Additionally, DELAG can quantify uncertainty that enhances LST reconstruction reliability. We further tested the reconstructed LST to estimate near-surface air temperature, achieving results (RMSE = 1.48-2.11 K) comparable to those derived from clear-sky LST (RMSE = 1.63-2.02 K). The results demonstrate the successful reconstruction through DELAG and highlight the broader applications of LST reconstruction for estimating accurate air temperature. Our study thus provides a novel and practical method for Landsat LST reconstruction, particularly suited for complex urban areas within Landsat cross-track areas, taking one step toward addressing complex climate events at high spatiotemporal resolution.
The aim of this paper is to develop novel quantum algorithms for Gaussian process quadrature methods. Gaussian process quadratures are numerical integration methods where Gaussian processes are used as functional priors for the integrands to capture the uncertainty arising from the sparse function evaluations. Quantum computers have emerged as potential replacements for classical computers, offering exponential reductions in the computational complexity of machine learning tasks. In this paper, we combine Gaussian process quadratures and quantum computing by proposing a quantum low-rank Gaussian process quadrature method based on a Hilbert space approximation of the Gaussian process kernel and enhancing the quadrature using a quantum circuit. The method combines the quantum phase estimation algorithm with the quantum principal component analysis technique to extract information up to a desired rank. Then, Hadamard and SWAP tests are implemented to find the expected value and variance that determines the quadrature. We use numerical simulations of a quantum computer to demonstrate the effectiveness of the method. Furthermore, we provide a theoretical complexity analysis that shows a polynomial advantage over classical Gaussian process quadrature methods. The code is available at https://github.com/cagalvisf/Quantum_HSGPQ.
Learning for control in repeated tasks allows for well-designed experiments to gather the most useful data. We consider the setting in which we use a data-driven controller that does not have access to the true system dynamics. Rather, the controller uses inferred dynamics based on the available information. In order to acquire data that is beneficial for this controller, we present an experimental design approach that leverages the current data to improve expected control performance. We focus on the setting in which inference on the unknown dynamics is performed using Gaussian processes. Gaussian processes not only provide uncertainty quantification but also allow us to leverage structures inherent to Gaussian random variables. Through this structure, we design experiments via gradient descent on the expected control performance with respect to the experiment input. In particular, we focus on a chance-constrained minimum expected time control problem. Numerical demonstrations of our approach indicate our experimental design outperforms relevant benchmarks.
To reduce the curse of dimensionality for Gaussian processes (GP), they can be decomposed into a Gaussian Process Network (GPN) of coupled subprocesses with lower dimensionality. In some cases, intermediate observations are available within the GPN. However, intermediate observations are often indirect, noisy, and incomplete in most real-world systems. This work introduces the Partially Observable Gaussian Process Network (POGPN) to model real-world process networks. We model a joint distribution of latent functions of subprocesses and make inferences using observations from all subprocesses. POGPN incorporates observation lenses (observation likelihoods) into the well-established inference method of deep Gaussian processes. We also introduce two training methods for POPGN to make inferences on the whole network using node observations. The application to benchmark problems demonstrates how incorporating partial observations during training and inference can improve the predictive performance of the overall network, offering a promising outlook for its practical application.
In the field of chemical engineering, traditional data-processing and prediction methods face significant challenges. Machine-learning and large-language models (LLMs) also have their respective limitations. This paper explores the application of the Chain-of-Thought (CoT) reasoning model in chemical engineering, starting from 30 experimental data points. By integrating traditional surrogate models like Gaussian processes and random forests with powerful LLMs such as DeepSeek-R1, a hierarchical architecture is proposed. Two CoT-building methods, Large Language Model-Chain of Thought (LLM-CoT) and Machine Learning-Large Language Model-Chain of Thought (ML-LLM-CoT), are studied. The LLM-CoT combines local models DeepSeek-r1:14b and Qwen2:7b with Ollama. The ML-LLM-CoT integrates a pre-trained Gaussian ML model with the LLM-based CoT framework. Our results show that during construction, ML-LLM-CoT is more efficient. It only has 2 points that require rethink and a total of 4 rethink times, while LLM-CoT has 5 points that need to be re-thought and 34 total rethink times. In predicting the solubility of 20 molecules with dissimilar structures, the number of molecules with a prediction deviation higher than 100\% for the Gaussian model, LLM-CoT, and ML-LLM-CoT is 7, 6, and 4 respectively. These results indicate that ML-LLM-CoT performs better in controlling the number of high-deviation molecules, optimizing the average deviation, and achieving a higher success rate in solubility judgment, providing a more reliable method for chemical engineering and molecular property prediction. This study breaks through the limitations of traditional methods and offers new solutions for rapid property prediction and process optimization in chemical engineering.
Bayesian Optimization (BO) for the minimization of expensive functions of continuous variables uses all the knowledge acquired from previous samples (${\boldsymbol x}_i$ and $f({\boldsymbol x}_i)$ values) to build a surrogate model based on Gaussian processes. The surrogate is then exploited to define the next point to sample, through a careful balance of exploration and exploitation. Initially intended for low-dimensional spaces, BO has recently been modified and used also for very large-dimensional spaces (up to about one thousand dimensions). In this paper we consider a much simpler algorithm, called "Reactive Affine Shaker" (RAS). The next sample is always generated with a uniform probability distribution inside a parallelepiped (the "box"). At each iteration, the form of the box is adapted during the search through an affine transformation, based only on the point $\boldsymbol x$ position and on the success or failure in improving the function. The function values are therefore not used directly to modify the search area and to generate the next sample. The entire dimensionality is kept (no active subspaces). Despite its extreme simplicity and its use of only stochastic local search, surprisingly the produced results are comparable to and not too far from the state-of-the-art results of high-dimensional versions of BO, although with some more function evaluations. An ablation study and an analysis of probability distribution of directions (improving steps and prevailing box orientation) in very large-dimensional spaces are conducted to understand more about the behavior of RAS and to assess the relative importance of the algorithmic building blocks for the final results.
With the strengths of both deep learning and kernel methods like Gaussian Processes (GPs), Deep Kernel Learning (DKL) has gained considerable attention in recent years. From the computational perspective, however, DKL becomes challenging when the input dimension of the GP layer is high. To address this challenge, we propose the Deep Additive Kernel (DAK) model, which incorporates i) an additive structure for the last-layer GP; and ii) induced prior approximation for each GP unit. This naturally leads to a last-layer Bayesian neural network (BNN) architecture. The proposed method enjoys the interpretability of DKL as well as the computational advantages of BNN. Empirical results show that the proposed approach outperforms state-of-the-art DKL methods in both regression and classification tasks.
Mean field games (MFGs) describe the collective behavior of large populations of interacting agents. In this work, we tackle ill-posed inverse problems in potential MFGs, aiming to recover the agents' population, momentum, and environmental setup from limited, noisy measurements and partial observations. These problems are ill-posed because multiple MFG configurations can explain the same data, or different parameters can yield nearly identical observations. Nonetheless, they remain crucial in practice for real-world scenarios where data are inherently sparse or noisy, or where the MFG structure is not fully determined. Our focus is on finding surrogate MFGs that accurately reproduce the observed data despite these challenges. We propose two Gaussian process (GP)-based frameworks: an inf-sup formulation and a bilevel approach. The choice between them depends on whether the unknown parameters introduce concavity in the objective. In the inf-sup framework, we use the linearity of GPs and their parameterization structure to maintain convex-concave properties, allowing us to apply standard convex optimization algorithms. In the bilevel framework, we employ a gradient-descent-based algorithm and introduce two methods for computing the outer gradient. The first method leverages an existing solver for the inner potential MFG and applies automatic differentiation, while the second adopts an adjoint-based strategy that computes the outer gradient independently of the inner solver. Our numerical experiments show that when sufficient prior information is available, the unknown parameters can be accurately recovered. Otherwise, if prior information is limited, the inverse problem is ill-posed, but our frameworks can still produce surrogate MFG models that closely match observed data.