For differential equations, differentiable integrators [see, e.g.,7]will constitute alayer.

Integrating physics models within machine learning models holds considerable promise toward learning robust models with improved interpretability and abilities to extrapolate. In this work, we focus on the integration of incomplete physics models into deep generative models. In particular, we introduce an architecture of variational autoencoders (VAEs) in which a part of the latent space is grounded by physics. A key technical challenge is to strike a balance between the incomplete physics and trainable components such as neural networks for ensuring that the physics part is used in a meaningful manner. To this end, we propose a regularized learning method that controls the effect of the trainable components and preserves the semantics of the physics-based latent variables as intended. We not only demonstrate generative performance improvements over a set of synthetic and real-world datasets, but we also show that we learn robust models that can consistently extrapolate beyond the training distribution in a meaningful manner. Moreover, we show that we can control the generative process in an interpretable manner.

The concept learner grounds visual concepts ( e.g., color, shape, and material) from these object-centric representations based on the language,

Intensive studies have been conducted in recent years to integrate neural networks with physics models to balance model accuracy and interpretability. One recently proposed approach, named Physics-Enhanced Residual Learning (PERL), is to use learning to estimate the residual between the physics model prediction and the ground truth. Numeral examples suggested that integrating such residual with physics models in PERL has three advantages: (1) a reduction in the number of required neural network parameters; (2) faster convergence rates; and (3) fewer training samples needed for the same computational precision. However, these numerical results lack theoretical justification and cannot be adequately explained. This paper aims to explain these advantages of PERL from a theoretical perspective. We investigate a general class of problems with Lipschitz continuity properties. By examining the relationships between the bounds to the loss function and residual learning structure, this study rigorously proves a set of theorems explaining the three advantages of PERL. Several numerical examples in the context of automated vehicle trajectory prediction are conducted to illustrate the proposed theorems. The results confirm that, even with significantly fewer training samples, PERL consistently achieves higher accuracy than a pure neural network. These results demonstrate the practical value of PERL in real world autonomous driving applications where corner case data are costly or hard to obtain. PERL therefore improves predictive performance while reducing the amount of data required.

A technical challenge in deep gray-box modeling is to ensure an appropriate use of physics models. A careless design of models and learning can lead to an erratic behavior of the components meant to represent physics (e.g., with erroneous estimation of physics parameters), and eventually, the overall

Quantifying and propagating modeling uncertainties is crucial for reliability analysis, robust optimization, and other model-based algorithmic processes in engineering design and control. Now, physics-informed machine learning (PIML) methods have emerged in recent years as a new alternative to traditional computational modeling and surrogate modeling methods, offering a balance between computing efficiency, modeling accuracy, and interpretability. However, their ability to predict and propagate modeling uncertainties remains mostly unexplored. In this paper, a promising class of auto-differentiable hybrid PIML architectures that combine partial physics and neural networks or ANNs (for input transformation or adaptive parameter estimation) is integrated with Bayesian Neural networks (replacing the ANNs); this is done with the goal to explore whether BNNs can successfully provision uncertainty propagation capabilities in the PIML architectures as well, further supported by the auto-differentiability of these architectures. A two-stage training process is used to alleviate the challenges traditionally encountered in training probabilistic ML models. The resulting BNN-integrated PIML architecture is evaluated on an analytical benchmark problem and flight experiments data for a fixed-wing RC aircraft, with prediction performance observed to be slightly worse or at par with purely data-driven ML and original PIML models. Moreover, Monte Carlo sampling of probabilistic BNN weights was found to be most effective in propagating uncertainty in the BNN-integrated PIML architectures.

Physics phenomena are often described by ordinary and/or partial differential equations (ODEs/PDEs), and solved analytically or numerically. Unfortunately, many real-world systems are described only approximately with missing or unknown terms in the equations. This makes the distribution of the physics model differ from the true data-generating process (DGP). Using limited and unpaired data between DGP observations and the imperfect model simulations, we investigate this particular setting by completing the known-physics model, combining theory-driven models and data-driven to describe the shifted distribution involved in the DGP. We present a novel hybrid generative model approach combining deep grey-box modelling with Optimal Transport (OT) methods to enhance incomplete physics models. Our method implements OT maps in data space while maintaining minimal source distribution distortion, demonstrating superior performance in resolving the unpaired problem and ensuring correct usage of physics parameters. Unlike black-box alternatives, our approach leverages physics-based inductive biases to accurately learn system dynamics while preserving interpretability through its domain knowledge foundation. Experimental results validate our method's effectiveness in both generation tasks and model transparency, offering detailed insights into learned physics dynamics.

Many inverse problems in nuclear fusion and high-energy astrophysics research, such as the optimization of tokamak reactor geometries or the inference of black hole parameters from interfer-ometric images, necessitate high-dimensional parameter scans and large ensembles of simulations to be performed. Such inverse problems typically involve large uncertainties, both in the measurement parameters being inverted and in the underlying physics models themselves. Monte Carlo sampling, when combined with modern non-linear dimensionality reduction techniques such as autoencoders and manifold learning, can be used to reduce the size of the parameter spaces considerably. However, there is no guarantee that the resulting combinations of parameters will be physically valid, or even mathematically consistent. In this position paper, we advocate adopting a hybrid approach that leverages our recent advances in the development of formal verification methods for numerical algorithms, with the goal of constructing parameter space restrictions with provable mathematical and physical correctness properties, whilst nevertheless respecting both experimental uncertainties and uncertainties in the underlying physical processes.