physics-informed neural processes

Physical models often suffer from systematic biases due to missing or misspecified mechanisms, and conventional calibration methods struggle to generalize efficiently across related systems. This work proposes the first population-level framework that integrates neural processes into physics-informed calibration. By combining differentiable physical modeling with a two-branch latent variable architecture, the approach enables amortized Bayesian inference across system instances, effectively disentangling instance-specific parameters from shared structural biases. The method facilitates rapid, uncertainty-aware calibration for previously unseen system instances and demonstrates substantially improved parameter recovery accuracy on benchmark tasks—including damped oscillators, Lotka–Volterra systems, and convection–diffusion partial differential equations—while correctly identifying the underlying bias structure.

This work proposes a variational Bayesian framework that jointly models the posterior and predictive distributions, circumventing the conventional two-stage pipeline of posterior approximation followed by Monte Carlo propagation. By leveraging amortized variational inference, the method completes training offline, thereby eliminating the need for repeated online sampling and significantly reducing computational overhead during prediction. The approach introduces a variational upper bound on the KL divergence together with a moment-matching regularization term to enable efficient and accurate uncertainty quantification. Evaluated on both analytical benchmarks and finite-element-based solid mechanics problems, the proposed method achieves substantially higher predictive accuracy compared to traditional approaches while drastically lowering online computational costs, making it well-suited for high-fidelity models where conventional Bayesian inference is prohibitively expensive.

This study investigates the statistical reliability, generalization capability, and uncertainty quantification performance of amortized Bayesian inference under varying signal-to-noise ratios and distributional shifts. By establishing the first systematic statistical connection between amortized inference and Bayesian computation, it analyzes how prevalent neural architectures—including feedforward networks, Deep Sets, and Transformers—enable structured probabilistic approximation and reveals the intrinsic mechanisms that support controllable generalization error. Empirical simulations demonstrate that amortized inference achieves high accuracy and robustness across diverse deployment scenarios, while also clarifying its strengths and limitations in uncertainty quantification.

Robots exhibit poor generalization, weak interpretability, and low data efficiency when adaptively learning in unknown dynamic environments. Method: This paper proposes a neuro-symbolic learning framework integrating physics-informed priors with data-driven learning. It unifies differentiable physics modeling, Bayesian uncertainty inference, and meta-learning within a single architecture. Specifically, dynamics constraints are encoded via a differentiable physics engine; cognitive uncertainty is quantified through Bayesian inverse problem solving; and neural-symbolic joint optimization, combined with meta-learning–driven few-shot task adaptation, enhances decision interpretability, cross-scenario transferability, and sample efficiency. Contribution/Results: Experiments demonstrate strong generalization under dynamic disturbances and out-of-distribution conditions, alongside robust continual learning capability. The framework establishes a reliable, transparent, and data-efficient learning paradigm for next-generation autonomous robots.

This work addresses the challenge of specifying prior distributions in Bayesian deep learning—particularly the difficulty of performing inference in the absence of initial beliefs—by proposing a learnable weight prior mechanism that, for the first time, integrates Bayesian neural networks with probabilistic meta-learning. Framed within the neural process paradigm, the approach treats network weights as latent variables and learns a shared prior across multiple tasks. Amortized variational inference is employed to efficiently approximate task-specific posteriors. The resulting model supports minibatch training within tasks and excels in meta-learning under extreme data scarcity. Beyond yielding more interpretable Bayesian neural network behavior through well-calibrated priors, the framework also functions as a flexible generative model, successfully accomplishing neural process tasks previously deemed infeasible.

This work addresses the lack of a unified understanding and reliable quantification of uncertainty in machine learning models applied to physical sciences, which undermines the credibility of scientific discovery. The authors propose a comprehensive uncertainty classification framework tailored to the intersection of physics and artificial intelligence, systematically clarifying the distinctions between predictive and inferential uncertainty from both Bayesian and frequentist perspectives. The framework integrates multidimensional validation techniques—including coverage, probabilistic calibration, bias diagnostics, and proper scoring rules—to rigorously assess uncertainty quality. Its effectiveness is demonstrated through representative regression and classification case studies, offering a structured and actionable guideline for evaluating uncertainty in AI-driven physical science applications.

This work addresses the challenge of Bayesian inference in Kuramoto oscillator networks, where high-dimensional state spaces and intractable likelihood functions hinder conventional approaches. To overcome this, the study introduces an amortized Bayesian inference framework that leverages neural networks to learn an approximation of the posterior distribution directly from simulated phase dynamics. By bypassing repeated sampling or iterative optimization, the method substantially enhances computational efficiency and enables practical uncertainty quantification. Experiments on synthetic Kuramoto networks demonstrate that the proposed approach accurately recovers the true posterior, effectively captures parameter uncertainty, and significantly reduces computational cost. This establishes a scalable, data-driven paradigm for Bayesian inference in complex dynamical systems.

Despite the strong predictive performance of deep neural networks, their generalization mechanisms and uncertainty quantification lack rigorous theoretical foundations. This work proposes a unified probabilistic framework that integrates Bayesian inference, function-space modeling, and large deviation theory, offering the first synthesis of diversity, smoothness, and stochasticity—the three key drivers of generalization—within the PAC-Bayes and large deviation paradigms. The core contributions include the Deep Variational Implicit Process (DVIP) model and two efficient post-hoc calibration methods, VaLLA and FMGP, which directly calibrate uncertainty estimates of pre-trained deterministic networks. These advances not only enable practical uncertainty quantification but also provide a theoretical explanation for the surprisingly good generalization observed in over-parameterized neural networks.

This study addresses the limitation of mean-field variational approximations in Gaussian Process Latent Variable Models (GPLVMs), which constrain manifold uncertainty estimation. To overcome this, we propose an amortized structured stochastic variational inference method. The core innovation lies in moving beyond the traditional mean-field assumption by leveraging neural network amortizers to model the conditional dependence between inducing points and latent variables. This enables the latent space posterior distribution to be conditioned on inducing points, thereby significantly enhancing model flexibility. Experimental results demonstrate that the proposed approach yields substantial improvements in both data manifold reconstruction metrics and the quality of uncertainty estimation. Consequently, this work establishes a superior structured Gaussian process inference paradigm for complex data representation.

This work proposes a novel Bayesian framework that explicitly incorporates linear equality constraints—encoding known physical laws—into variational Bayesian inference, thereby jointly optimizing physical consistency and uncertainty quantification. By integrating Bayesian neural networks with a constraint-embedding mechanism, the method enables unified uncertainty modeling over both model parameters and domain knowledge. Evaluated on a single-particle battery modeling task, the approach significantly narrows predictive credible intervals and drastically reduces violations of the prescribed linear constraints compared to standard variational Bayesian neural networks, demonstrating its effectiveness and practical utility in delivering reliable, physics-informed predictions with well-calibrated uncertainty estimates.