Allan P. Engsig-Karup

Daniele Bigoni, Allan P. Engsig-Karup, Youssef M. Marzouk

The accurate approximation of high-dimensional functions is an essential task in uncertainty quantification and many other fields. We propose a new function approximation scheme based on a spectral extension of the tensor-train (TT) decomposition. We first define a functional version of the TT decomposition and analyze its properties. We obtain results on the convergence of the decomposition, revealing links between the regularity of the function, the dimension of the input space, and the TT ranks. We also show that the regularity of the target function is preserved by the univariate functions (i.e., the "cores") comprising the functional TT decomposition. This result motivates an approximation scheme employing polynomial approximations of the cores. For functions with appropriate regularity, the resulting \textit{spectral tensor-train decomposition} combines the favorable dimension-scaling of the TT decomposition with the spectral convergence rate of polynomial approximations, yielding efficient and accurate surrogates for high-dimensional functions. To construct these decompositions, we use the sampling algorithm \texttt{TT-DMRG-cross} to obtain the TT decomposition of tensors resulting from suitable discretizations of the target function. We assess the performance of the method on a range of numerical examples: a modifed set of Genz functions with dimension up to $100$, and functions with mixed Fourier modes or with local features. We observe significant improvements in performance over an anisotropic adaptive Smolyak approach. The method is also used to approximate the solution of an elliptic PDE with random input data. The open source software and examples presented in this work are available online.

Nikolas Borrel-Jensen, Somdatta Goswami, Allan P. Engsig-Karup et al.

We address the challenge of sound propagation simulations in 3D virtual rooms with moving sources, which have applications in virtual/augmented reality, game audio, and spatial computing. Solutions to the wave equation can describe wave phenomena such as diffraction and interference. However, simulating them using conventional numerical discretization methods with hundreds of source and receiver positions is intractable, making stimulating a sound field with moving sources impractical. To overcome this limitation, we propose using deep operator networks to approximate linear wave-equation operators. This enables the rapid prediction of sound propagation in realistic 3D acoustic scenes with moving sources, achieving millisecond-scale computations. By learning a compact surrogate model, we avoid the offline calculation and storage of impulse responses for all relevant source/listener pairs. Our experiments, including various complex scene geometries, show good agreement with reference solutions, with root mean squared errors ranging from 0.02 Pa to 0.10 Pa. Notably, our method signifies a paradigm shift as no prior machine learning approach has achieved precise predictions of complete wave fields within realistic domains. We anticipate that our findings will drive further exploration of deep neural operator methods, advancing research in immersive user experiences within virtual environments.$

Freja Høgholm Petersen, Jesper Sandvig Mariegaard, Rocco Palmitessa et al.

While POD-based surrogates are widely explored for hydrodynamic applications, the use of Koopman Autoencoders for real-world coastal-ocean modelling remains relatively limited. This paper introduces a flexible Koopman autoencoder formulation that incorporates meteorological forcings and boundary conditions, and systematically compares its performance against POD-based surrogates. The Koopman autoencoder employs a learned linear temporal operator in latent space, enabling eigenvalue regularization to promote temporal stability. This strategy is evaluated alongside temporal unrolling techniques for achieving stable and accurate long-term predictions. The models are assessed on three test cases spanning distinct dynamical regimes, with prediction horizons up to one year at 30-minute temporal resolution. Across all cases, the Koopman autoencoder with temporal unrolling yields the best overall accuracy compared to the POD-based surrogates, achieving relative root-mean-squared-errors of 0.01-0.13 and $R^2$-values of 0.65-0.996. Prediction errors are largest for current velocities, and smallest for water surface elevations. Comparing to in-situ observations, the surrogate yields -0.65% to 12% change in water surface elevation prediction error when compared to prediction errors of the physics-based model. These error levels, corresponding to a few centimeters, are acceptable for many practical applications, while inference speed-ups of 300-1400x enables workflows such as ensemble forecasting and long climate simulations for coastal-ocean modelling.

Yuanxin Xia, Xinyan Li, Matteo Calafà et al.

Standardized laboratory characterizations for absorbing materials rely on idealized sound field assumptions, which deviate largely from real-life conditions. Consequently, \emph{in-situ} acoustic characterization has become essential for accurate diagnosis and virtual prototyping. We propose a physics-informed neural field that reconstructs local, near-surface broadband sound fields from sparse pressure samples to directly infer complex surface impedance. A parallel, multi-frequency architecture enables a broadband impedance retrieval within runtimes on the order of seconds to minutes. To validate the method, we developed a compact microphone array with low hardware complexity. Numerical verifications and laboratory experiments demonstrate accurate impedance retrieval with a small number of sensors under realistic conditions. We further showcase the approach in a vehicle cabin to provide practical guidance on measurement locations that avoid strong interference. Here, we show that this approach offers a robust means of characterizing \emph{in-situ} boundary conditions for architectural and automotive acoustics.

Nikolas Borrel-Jensen, Allan P. Engsig-Karup, Cheol-Ho Jeong

Realistic sound is essential in virtual environments, such as computer games and mixed reality. Efficient and accurate numerical methods for pre-calculating acoustics have been developed over the last decade; however, pre-calculating acoustics makes handling dynamic scenes with moving sources challenging, requiring intractable memory storage. A physics-informed neural network (PINN) method in 1D is presented, which learns a compact and efficient surrogate model with parameterized moving Gaussian sources and impedance boundaries, and satisfies a system of coupled equations. The model shows relative mean errors below 2%/0.2 dB and proposes a first step in developing PINNs for realistic 3D scenes.

Hermes Sampedro Llopis, Allan P. Engsig-Karup, Cheol-Ho Jeong et al.

The use of model-based numerical simulation of wave propagation in rooms for engineering applications requires that acoustic conditions for multiple parameters are evaluated iteratively and this is computationally expensive. We present a reduced basis methods (RBM) to achieve a computational cost reduction relative to a traditional full order model (FOM), for wave-based room acoustic simulations with parametrized boundary conditions. In this study, the FOM solver is based on the spectral element method, however other numerical methods could be applied. The RBM reduces the computational burden by solving the problem in a low-dimensional subspace for parametrized frequency-independent and frequency-dependent boundary conditions. The problem is formulated and solved in the Laplace domain, which ensures the stability of the reduced order model based on the RBM approach. We study the potential of the proposed RBM framework in terms of computational efficiency, accuracy and storage requirements and we show that the RBM leads to 100-fold speed-ups for a 2D case with an upper frequency of 2kHz and around 1000-fold speed-ups for an analogous 3D case with an upper frequency of 1kHz. While the FOM simulations needed to construct the ROM are expensive, we demonstrate that despite this cost, the ROM has a potential of three orders of magnitude faster than the FOM when four different boundary conditions are simulated per room surface. Moreover, results show that the storage model for the ROM is relatively high but affordable for the presented 2D and 3D cases.