Transferring the recent advancements in deep learning into scientific disciplines is hindered by the lack of the required large-scale datasets for training. We argue that in these knowledge-rich domains, the established body of scientific theory provides reliable inductive biases in the form of governing physical laws. We address the ill-posed inverse problem of recovering Raman spectra from noisy Coherent Anti-Stokes Raman Scattering (CARS) measurements, as the true Raman signal here is suppressed by a dominating non-resonant background. We propose RamPINN, a model that learns to recover Raman spectra from given CARS spectra. Our core methodological contribution is a physics-informed neural network that utilizes a dual-decoder architecture to disentangle resonant and non-resonant signals. This is done by enforcing the Kramers-Kronig causality relations via a differentiable Hilbert transform loss on the resonant and a smoothness prior on the non-resonant part of the signal. Trained entirely on synthetic data, RamPINN demonstrates strong zero-shot generalization to real-world experimental data, explicitly closing this gap and significantly outperforming existing baselines. Furthermore, we show that training with these physics-based losses alone, without access to any ground-truth Raman spectra, still yields competitive results. This work highlights a broader concept: formal scientific rules can act as a potent inductive bias, enabling robust, self-supervised learning in data-limited scientific domains.

We introduce the Divergence Phase Index (DPI), a novel framework for quantifying phase differences in one and multidimensional signals, grounded in harmonic analysis via the Riesz transform. Based on classical Hilbert Transform phase measures, the DPI extends these principles to higher dimensions, offering a geometry-aware metric that is invariant to intensity scaling and sensitive to structural changes. We applied this method on both synthetic and real-world datasets, including intracranial EEG (iEEG) recordings during epileptic seizures, high-resolution microscopy images, and paintings. In the 1D case, the DPI robustly detects hypersynchronization associated with generalized epilepsy, while in 2D, it reveals subtle, imperceptible changes in images and artworks. Additionally, it can detect rotational variations in highly isotropic microscopy images. The DPI's robustness to amplitude variations and its adaptability across domains enable its use in diverse applications from nonlinear dynamics, complex systems analysis, to multidimensional signal processing.

Neural operators have emerged as a powerful, data-driven paradigm for learning solution operators of partial differential equations (PDEs). State-of-the-art architectures, such as the Fourier Neural Operator (FNO), have achieved remarkable success by performing convolutions in the frequency domain, making them highly effective for a wide range of problems. However, this method has some limitations, including the periodicity assumption of the Fourier transform. In addition, there are other methods of analysing a signal, beyond phase and amplitude perspective, and provide us with other useful information to learn an effective network. We introduce the \textbf{Hilbert Neural Operator (HNO)}, a new neural operator architecture to address some advantages by incorporating a strong inductive bias from signal processing. HNO operates by first mapping the input signal to its analytic representation via the Hilbert transform, thereby making instantaneous amplitude and phase information explicit features for the learning process. The core learnable operation -- a spectral convolution -- is then applied to this Hilbert-transformed representation. We hypothesize that this architecture enables HNO to model operators more effectively for causal, phase-sensitive, and non-stationary systems. We formalize the HNO architecture and provide the theoretical motivation for its design, rooted in analytic signal theory.

Toeplitz Neural Networks (TNNs) (Qin et. al. 2023) are a recent sequence model with impressive results. They require O(n log n) computational complexity and O(n) relative positional encoder (RPE) multi-layer perceptron (MLP) and decay bias calls. We aim to reduce both. We first note that the RPE is a non-SPD (symmetric positive definite) kernel and the Toeplitz matrices are pseudo-Gram matrices. Further 1) the learned kernels display spiky behavior near the main diagonals with otherwise smooth behavior; 2) the RPE MLP is slow. For bidirectional models, this motivates a sparse plus low-rank Toeplitz matrix decomposition. For the sparse component's action, we do a small 1D convolution. For the low rank component, we replace the RPE MLP with linear interpolation and use asymmetric Structured Kernel Interpolation (SKI) (Wilson et. al. 2015) for O(n) complexity: we provide rigorous error analysis. For causal models, "fast" causal masking (Katharopoulos et. al. 2020) negates SKI's benefits. Working in the frequency domain, we avoid an explicit decay bias. To enforce causality, we represent the kernel via the real part of its frequency response using the RPE and compute the imaginary part via a Hilbert transform. This maintains O(n log n) complexity but achieves an absolute speedup. Modeling the frequency response directly is also competitive for bidirectional training, using one fewer FFT. We set a speed state of the art on Long Range Arena (Tay et. al. 2020) with minimal score degradation.

Department of Mathematics, Princeton University, Princeton, NJ 08544, USA (Dated: May 9, 2023) Whether there exist finite time blow-up solutions for the 2-D Boussinesq and the 3-D Euler equations are of fundamental importance to the field of fluid mechanics. We develop a new numerical framework, employing physics-informed neural networks (PINNs), that discover, for the first time, a smooth self-similar blow-up profile for both equations. The solution itself could form the basis of a future computer-assisted proof of blow-up for both equations. In addition, we demonstrate PINNs could be successfully applied to find unstable self-similar solutions to fluid equations by constructing the first example of an unstable self-similar solution to the Córdoba-Córdoba-Fontelos equation. We show that our numerical framework is both robust and adaptable to various other equations. A celebrated open question in fluids is whether or not cylindrical boundary is intrinsically linked to the same from smooth initial data the 3-D Euler equations may problem for the 2-D Bousinessq equations (cf. The mechanism for blowup self-similar blow-up was proven in the groundbreaking for the two equations is believed to be identical.

Several new properties of weighted Hilbert transform are obtained. If mu is zero, two Plancherel-like equations and the isotropic properties are derived. For mu is real number, a coerciveness is derived and two iterative sequences are constructed to find the inversion. The proposed iterative sequences are applicable to the case of pure imaginary constant mu=i*eta with |eta|

This revisit gives a survey on the analytical methods for the inverse exponential Radon transform which has been investigated in the past three decades from both mathematical interests and medical applications such as nuclear medicine emission imaging. The derivation of the classical inversion formula is through the recent argument developed for the inverse attenuated Radon transform. That derivation allows the exponential parameter to be a complex constant, which is useful to other applications such as magnetic resonance imaging and tensor field imaging. The survey also includes the new technique of using the finite Hilbert transform to handle the exact reconstruction from 180 degree data. Special treatment has been paid on two practically important subjects. One is the exact reconstruction from partial measurements such as half-scan and truncated-scan data, and the other is the reconstruction from diverging-beam data. The noise propagation in the reconstruction is touched upon with more heuristic discussions than mathematical inference. The numerical realizations of several classical reconstruction algorithms are included. In the conclusion, several topics are discussed for more investigations in the future.

Deep learning has become an area of interest in most scientific areas, including physical sciences. Modern networks apply real-valued transformations on the data. Particularly, convolutions in convolutional neural networks discard phase information entirely. Many deterministic signals, such as seismic data or electrical signals, contain significant information in the phase of the signal. We explore complex-valued deep convolutional networks to leverage non-linear feature maps. Seismic data commonly has a lowcut filter applied, to attenuate noise from ocean waves and similar long wavelength contributions. Discarding the phase information leads to low-frequency aliasing analogous to the Nyquist-Shannon theorem for high frequencies. In non-stationary data, the phase content can stabilize training and improve the generalizability of neural networks. While it has been shown that phase content can be restored in deep neural networks, we show how including phase information in feature maps improves both training and inference from deterministic physical data. Furthermore, we show that the reduction of parameters in a complex network results in training on a smaller dataset without overfitting, in comparison to a real-valued network with the same performance.

Computed tomography for region-of-interest (ROI) reconstruction has advantages of reducing X-ray radiation dose and using a small detector. However, standard analytic reconstruction methods suffer from severe cupping artifacts, and existing model-based iterative reconstruction methods require extensive computations. Recently, we proposed a deep neural network to learn the cupping artifact, but the network is not well generalized for different ROIs due to the singularities in the corrupted images. Therefore, there is an increasing demand for a neural network that works well for any ROI sizes. In this paper, two types of neural networks are designed. The first type learns ROI size-specific cupping artifacts from the analytic reconstruction images, whereas the second type network is to learn to invert the finite Hilbert transform from the truncated differentiated backprojection (DBP) data. Their generalizability for any ROI sizes is then examined. Experimental results show that the new type of neural network significantly outperforms the existing iterative methods for any ROI size in spite of significantly reduced run-time complexity. Since the proposed method consistently surpasses existing methods for any ROIs, it can be used as a general CT reconstruction engine for many practical applications without compromising possible detector truncation.