Steven H. Frankel

Shimon Pisnoy, Hemanth Chandravamsi, Ziv Chen et al.

We present PINNACLE, an open-source computational framework for physics-informed neural networks (PINNs) that integrates modern training strategies, multi-GPU acceleration, and hybrid quantum-classical architectures within a unified modular workflow. The framework enables systematic evaluation of PINN performance across benchmark problems including 1D hyperbolic conservation laws, incompressible flows, and electromagnetic wave propagation. It supports a range of architectural and training enhancements, including Fourier feature embeddings, random weight factorization, strict boundary condition enforcement, adaptive loss balancing, curriculum training, and second-order optimization strategies, with extensibility to additional methods. We provide a comprehensive benchmark study quantifying the impact of these methods on convergence, accuracy, and computational cost, and analyze distributed data parallel scaling in terms of runtime and memory efficiency. In addition, we extend the framework to hybrid quantum-classical PINNs and derive a formal estimate for circuit-evaluation complexity under parameter-shift differentiation. Results highlight the sensitivity of PINNs to architectural and training choices, confirm their high computational cost relative to classical solvers, and identify regimes where hybrid quantum models offer improved parameter efficiency. PINNACLE provides a foundation for benchmarking physics-informed learning methods and guiding future developments through quantitative assessment of their trade-offs.

Dhanush Vittal Shenoy, Steven H. Frankel

Representing turbulent flow fields in a compact yet physically faithful form remains a central challenge in computational fluid dynamics. We propose a continuous parametric representation based on localized Gaussian primitives, in which the velocity field is modeled as a superposition of kernels with learnable positions, amplitudes, and scales. This formulation yields a compact, grid-independent encoding while enabling evaluation of derived quantities such as vorticity and enstrophy. The approach is assessed on three-dimensional Taylor-Green vortex fields spanning stages from smooth flow to fully developed turbulence. We quantify the compression-accuracy trade-off using both primary variables and derivative-sensitive diagnostics. The baseline isotropic formulation achieves high velocity accuracy at compression ratios exceeding 1e3-1e4, but exhibits substantial enstrophy degradation due to loss of small-scale structure. To address this limitation, we investigate structure-aware extensions including adaptive placement, multi-resolution kernels, and anisotropic Gaussians. The anisotropic formulation provides the most consistent improvement, better aligning with elongated vortical structures and recovering intermediate- and high-wavenumber content, while other strategies yield modest gains. A compact-support Beta basis improves enstrophy in some cases but introduces localized artifacts. Overall, the results indicate that the main limitation of baseline Gaussian representations lies in geometric expressiveness rather than parameter count. The proposed framework provides a compact, interpretable, and continuous representation of turbulent flows, and establishes a foundation for structure-aware and physics-informed flow compression.

Hemanth Chandravamsi, Dhanush V. Shenoy, Itay Zinn et al.

Deep neural networks are known to exhibit a spectral learning bias, wherein low-frequency components are learned early in training, while high-frequency modes emerge more gradually in later epochs. However, when the target signal lacks low-frequency components and is dominated by broadband high frequencies, training suffers from a 'spectral bottleneck', and the model fails to reconstruct the entire signal, including the frequency components that lie within the network's representational capacity. We examine such a scenario in the context of implicit neural representations (INRs) with sinusoidal representation networks (SIRENs), focusing on the challenge of fitting high-frequency-dominant signals that are susceptible to spectral bottleneck. To effectively fit any target signal irrespective of it's frequency content, we propose a generalized target-aware 'weight perturbation scheme' (WINNER - weight initialization with noise for neural representations) for network initialization. The scheme perturbs uniformly initialized weights with Gaussian noise, where the noise scales are adaptively determined by the spectral centroid of the target signal. We show that the noise scales can provide control over the spectra of network activations and the eigenbasis of the empirical neural tangent kernel. This method not only addresses the spectral bottleneck but also yields faster convergence and with improved representation accuracy, outperforming state-of-the-art approaches in audio fitting and achieving notable gains in image fitting and denoising tasks. Beyond signal reconstruction, our approach opens new directions for adaptive weight initialization strategies in computer vision and scientific machine learning.

Hemanth Chandravamsi, Dhanush V. Shenoy, Steven H. Frankel

We identify and address a fundamental limitation of sinusoidal representation networks (SIRENs), a class of implicit neural representations. SIRENs Sitzmann et al. (2020), when not initialized appropriately, can struggle at fitting signals that fall outside their frequency support. In extreme cases, when the network's frequency support misaligns with the target spectrum, a 'spectral bottleneck' phenomenon is observed, where the model yields to a near-zero output and fails to recover even the frequency components that are within its representational capacity. To overcome this, we propose WINNER - Weight Initialization with Noise for Neural Representations. WINNER perturbs uniformly initialized weights of base SIREN with Gaussian noise - whose noise scales are adaptively determined by the spectral centroid of the target signal. Similar to random Fourier embeddings, this mitigates 'spectral bias' but without introducing additional trainable parameters. Our method achieves state-of-the-art audio fitting and significant gains in image and 3D shape fitting tasks over base SIREN. Beyond signal fitting, WINNER suggests new avenues in adaptive, target-aware initialization strategies for optimizing deep neural network training. For code and data visit cfdlabtechnion.github.io/siren_square/.

Joao Felipe Gueiros, Hemanth Chandravamsi, Steven H. Frankel

This paper explores the use of deep residual networks for pricing European options on Petrobras, one of the world's largest oil and gas producers, and compares its performance with the Black-Scholes (BS) model. Using eight years of historical data from B3 (Brazilian Stock Exchange) collected via web scraping, a deep learning model was trained using a custom built hybrid loss function that incorporates market data and analytical pricing. The data for training and testing were drawn between the period spanning November 2016 to January 2025, using an 80-20 train-test split. The test set consisted of data from the final three months: November, December, and January 2025. The deep residual network model achieved a 64.3\% reduction in the mean absolute error for the 3-19 BRL (Brazilian Real) range when compared to the Black-Scholes model on the test set. Furthermore, unlike the Black-Scholes solution, which tends to decrease its accuracy for longer periods of time, the deep learning model performed accurately for longer expiration periods. These findings highlight the potential of deep learning in financial modeling, with future work focusing on specialized models for different price ranges.