In this paper, we propose a novel data-driven framework for discovering probabilistic laws underlying the Feynman-Kac formula. Specifically, we introduce the first stochastic SINDy method formulated under the risk-neutral probability measure to recover the backward stochastic differential equation (BSDE) from a single pair of stock and option trajectories. Unlike existing approaches to identifying stochastic differential equations-which typically require ergodicity-our framework leverages the risk-neutral measure, thereby eliminating the ergodicity assumption and enabling BSDE recovery from limited financial time series data. Using this algorithm, we are able not only to make forward-looking predictions but also to generate new synthetic data paths consistent with the underlying probabilistic law.

Physics-informed neural operators offer a powerful framework for learning solution operators of partial differential equations (PDEs) by combining data and physics losses. However, these physics losses rely on derivatives. Computing these derivatives remains challenging, with spectral and finite difference methods introducing approximation errors due to finite resolution. Here, we propose the mollified graph neural operator ($m$GNO), the first method to leverage automatic differentiation and compute exact gradients on arbitrary geometries. This enhancement enables efficient training on irregular grids and varying geometries while allowing seamless evaluation of physics losses at randomly sampled points for improved generalization. For a PDE example on regular grids, $m$GNO paired with autograd reduced the L2 relative data error by 20x compared to finite differences, although training was slower. It can also solve PDEs on unstructured point clouds seamlessly, using physics losses only, at resolutions vastly lower than those needed for finite differences to be accurate enough. On these unstructured point clouds, $m$GNO leads to errors that are consistently 2 orders of magnitude lower than machine learning baselines (Meta-PDE, which accelerates PINNs) for comparable runtimes, and also delivers speedups from 1 to 3 orders of magnitude compared to the numerical solver for similar accuracy. $m$GNOs can also be used to solve inverse design and shape optimization problems on complex geometries.

Although Finite Element Analysis (FEA) is an integral part of the product design lifecycle, the analysis is computationally expensive, making it unsuitable for many design optimization problems. The deep learning models can be a great solution. However, selecting the architecture that emulates the FEA with great accuracy is a challenge. This paper presents a comprehensive evaluation of graph neural networks (GNNs) and 3D U-Nets as surrogates for FEA of parametric I-beams. We introduce a Physics-Informed Neural Network (PINN) framework, governed by the Navier Cauchy equations, to enforce physical laws. Crucially, we demonstrate that a curriculum learning strategy, pretraining on data followed by physics informed fine tuning, is essential for stabilizing training. Our results show that GNNs fundamentally outperform the U-Net. Even the worst performer among GNNs, the GCN framework, achieved a relative L2 error of 8.7% while the best framework among U Net, U Net with attention mechanism trained on high resolution data, achieved 13.0% score. Among the graph-based architectures, the Message Passing Neural Networks (MPNN) and Graph Transformers achieved the highest accuracy, achieving a relative L2 score of 3.5% and 2.6% respectively. The inclusion of physics fundamental laws (PINN) significantly improved the generalization, reducing error by up to 11.3% on high-signal tasks. While the Graph Transformer is the most accurate model, it is more 37.5% slower during inference when compared to second best model, MPNN PINN. The PINN enhanced MPNN (MPNN PINN) provides the most practical solution. It offers a good compromise between predictive performance, model size, and inference speed.

Approximating solutions to partial differential equations (PDEs) is fundamental for the modeling of dynamical systems in science and engineering. Physics-informed neural networks (PINNs) are a recent machine learning-based approach, for which many properties and limitations remain unknown. PINNs are widely accepted as inferior to traditional methods for solving PDEs, such as the finite element method, both with regard to computation time and accuracy. However, PINNs are commonly claimed to show promise in solving inverse problems and handling noisy or incomplete data. We compare the performance of PINNs in solving inverse problems with that of a traditional approach using the finite element method combined with a numerical optimizer. The models are tested on a series of increasingly difficult fluid mechanics problems, with and without noise. We find that while PINNs may require less human effort and specialized knowledge, they are outperformed by the traditional approach. However, the difference appears to decrease with higher dimensions and more data. We identify common failures during training to be addressed if the performance of PINNs on noisy inverse problems is to become more competitive.

ABSTRACT It was recently shown that the loss function used for training physics-informed neural networks (PINNs) exhibits local minima at solutions corresponding to fixed points of dynamical systems. In the forward setting, where the PINN is trained to solve initial value problems, these local minima can interfere with training and potentially leading to physically incorrect solutions. Building on stability theory, this paper proposes a regularization scheme that penalizes solutions corresponding to unstable fixed points. Experimental results on four dynamical systems, including the Lotka-V olterra model and the van der Pol oscillator, show that our scheme helps avoiding physically incorrect solutions and substantially improves the training success rate of PINNs. Index T erms-- PINNs, regularization, stability 1. INTRODUCTION Physics-informed neural networks (PINNs, [1]) are among the most prominent instantiations of physics-informed machine learning.

Photoacoustic computed tomography (PACT) combines optical contrast with ultrasonic resolution, achieving deep-tissue imaging beyond the optical diffusion limit. While three-dimensional PACT systems enable high-resolution volumetric imaging for applications spanning transcranial to breast imaging, current implementations require dense transducer arrays and prolonged acquisition times, limiting clinical translation. We introduce Pano (PACT imaging neural operator), an end-to-end physics-aware model that directly learns the inverse acoustic mapping from sensor measurements to volumetric reconstructions. Unlike existing approaches (e.g. universal back-projection algorithm), Pano learns both physics and data priors while also being agnostic to the input data resolution. Pano employs spherical discrete-continuous convolutions to preserve hemispherical sensor geometry, incorporates Helmholtz equation constraints to ensure physical consistency and operates resolutionindependently across varying sensor configurations. We demonstrate the robustness and efficiency of Pano in reconstructing high-quality images from both simulated and real experimental data, achieving consistent performance even with significantly reduced transducer counts and limited-angle acquisition configurations. The framework maintains reconstruction fidelity across diverse sparse sampling patterns while enabling real-time volumetric imaging capabilities. This advancement establishes a practical pathway for making 3D PACT more accessible and feasible for both preclinical research and clinical applications, substantially reducing hardware requirements without compromising image reconstruction quality.

In the rapidly evolving domain of autonomous systems, interaction among agents within a shared environment is both inevitable and essential for enhancing overall system capabilities. A key requirement in such multi-agent systems is the ability of each agent to reliably predict the future positions of its nearest neighbors. Traditionally, graphs and graph theory have served as effective tools for modeling inter agent communication and relationships. While this approach is widely used, the present work proposes a novel method that leverages dynamic graphs in a forward looking manner. Specifically, the employment of EvolveGCN, a dynamic graph convolutional network, to forecast the evolution of inter-agent relationships over time. To improve prediction accuracy and ensure physical plausibility, this research incorporates physics constrained loss functions based on the Clohessy-Wiltshire equations of motion. This integrated approach enhances the reliability of future state estimations in multi-agent scenarios.

We present a robust neural watermarking framework for scientific data integrity, targeting high-dimensional fields common in climate modeling and fluid simulations. Using a convolutional autoencoder, binary messages are invisibly embedded into structured data such as temperature, vorticity, and geopotential. Our method ensures watermark persistence under lossy transformations - including noise injection, cropping, and compression - while maintaining near-original fidelity (sub-1\% MSE). Compared to classical singular value decomposition (SVD)-based watermarking, our approach achieves $>$98\% bit accuracy and visually indistinguishable reconstructions across ERA5 and Navier-Stokes datasets. This system offers a scalable, model-compatible tool for data provenance, auditability, and traceability in high-performance scientific workflows, and contributes to the broader goal of securing AI systems through verifiable, physics-aware watermarking. We evaluate on physically grounded scientific datasets as a representative stress-test; the framework extends naturally to other structured domains such as satellite imagery and autonomous-vehicle perception streams.

3D reconstruction of highly deformable surfaces (e.g. cloths) from monocular RGB videos is a challenging problem, and no solution provides a consistent and accurate recovery of fine-grained surface details. To account for the ill-posed nature of the setting, existing methods use deformation models with statistical, neural, or physical priors. They also predominantly rely on nonadaptive discrete surface representations (e.g. polygonal meshes), perform frame-by-frame optimisation leading to error propagation, and suffer from poor gradients of the mesh-based differentiable renderers. Consequently, fine surface details such as cloth wrinkles are often not recovered with the desired accuracy. In response to these limitations, we propose ThinShell-SfT, a new method for non-rigid 3D tracking that represents a surface as an implicit and continuous spatiotemporal neural field. We incorporate continuous thin shell physics prior based on the Kirchhoff-Love model for spatial regularisation, which starkly contrasts the discretised alternatives of earlier works. Lastly, we leverage 3D Gaussian splatting to differentiably render the surface into image space and optimise the deformations based on analysis-bysynthesis principles. Our Thin-Shell-SfT outperforms prior works qualitatively and quantitatively thanks to our continuous surface formulation in conjunction with a specially tailored simulation prior and surface-induced 3D Gaussians. See our project page at https://4dqv.mpiinf.mpg.de/ThinShellSfT.

Estimating the evolution of the battery's State of Charge (SoC) in response to its usage is critical for implementing effective power management policies and for ultimately improving the system's lifetime. Most existing estimation methods are either physics-based digital twins of the battery or data-driven models such as Neural Networks (NNs). In this work, we propose two new contributions in this domain. First, we introduce a novel NN architecture formed by two cascaded branches: one to predict the current SoC based on sensor readings, and one to estimate the SoC at a future time as a function of the load behavior. Second, we integrate battery dynamics equations into the training of our NN, merging the physics-based and data-driven approaches, to improve the models' generalization over variable prediction horizons. We validate our approach on two publicly accessible datasets, showing that our Physics-Informed Neural Networks (PINNs) outperform purely data-driven ones while also obtaining superior prediction accuracy with a smaller architecture with respect to the state-of-the-art.