iPINN for Broadband CARS Phase Retrieval: A Framework for Function Approximation and Inverse Modeling Problems in Nonlinear Spectroscopy

Phase retrieval in broadband coherent anti-Stokes Raman spectroscopy (BCARS) is an ill-posed inverse problem. The Raman-like signal is encoded in the imaginary part of the resonant susceptibility, which mixes coherently with a non-resonant background (NRB) that varies across acquisitions. We introduce an inverse physics-informed neural network (iPINN) that predicts Lorentzian peak parameters from raw BCARS spectra and reconstructs the resonant susceptibility through a differentiable analytical forward model. A transformer encoder assigns spectral features to 24 learnable peak slots, and a multi-view consistency loss enforces invariance across NRB pattern, NRB strength, and noise. Unlike direct spectral regression approaches, the method retains accuracy under varying acquisition conditions. On a public benchmark, iPINN achieves the lowest error among the tested baselines (MAE 0.016 vs. next-best 0.046). On 28 zero-shot test spectra acquired across seven solvents and four focal positions, accuracy is depth-invariant in five of seven solvents. These results show that inverse parametric prediction with a differentiable physical decoder supports robust phase retrieval across measurement conditions.

Harnessing Magnitude-Only and Complex Measurements for Improved Dynamic MRI Reconstruction with Learned Priors

MRI reconstruction methods for undersampled k-space data naturally utilize complex-valued measurements. Parallel developments in sparse phase retrieval have shown that magnitude-only measurements may provide complementary information for signal recovery. However, their use in MRI reconstruction remains largely unexplored, due to lack of practical settings where informative magnitude measurements can be obtained without additional scan time. In this work, we investigate the use of auxiliary k-space magnitude information for accelerated steady-state dynamic MRI reconstruction, and demonstrate strong consistency of k-space magnitudes across time-frames. Building on this observation, we propose $\mathbb{C}+\text{Mag}$, a magnitude-informed physics-driven deep learning reconstruction method. The proposed method employs an ADMM-based unrolling framework with a novel magnitude-aware data-fidelity formulation, where quadratically smoothed optimization and momentum-based updates are introduced to address the non-differentiability and non-convexity of the magnitude constraints. Experiments on retrospectively undersampled cine MRI and phase-contrast flow MRI datasets, as well as prospectively undersampled real-time cine MRI acquisitions, demonstrate improved artifact suppression, sharper anatomical recovery, and better preservation of phase information compared to conventional PD-DL methods, which is further supported through blinded expert reader evaluations.

Approximate Message Passing with Random Initialization for Phase Retrieval

We analyze approximate message passing (AMP) with an independent Gaussian initialization for noiseless phase retrieval in the proportional asymptotic regime. A random initialization has overlap of order $d^{-1/2}$ with the signal, and AMP requires a growing number of iterations to attain non-vanishing overlap. Thus, its precise behavior cannot be characterized by classical fixed-time state evolution. We prove a Gaussian decomposition of the AMP trajectory and control its error over the horizons required for recovery. The resulting analysis shows that random initialization attains the weak-recovery threshold $δ_{\rm weak}=1/2$. For $δ\in(δ_{\rm weak},δ_{\rm str})$, where $δ_{\rm str}\approx1.13$, the signal strength follows state evolution and approaches its stable finite fixed point uniformly for \(n^{1/3}/\operatorname{polylog}(n)\) iterations. For $δ>δ_{\rm str}$, AMP reaches any prescribed fixed recovery accuracy within $O_{δ,\varepsilon}(\log n)$ iterations. The majority of our analysis applies more generally to generalized AMP for single-index models.

On the conditional equivalence of phase retrieval algorithms

Phase retrieval - recovering a complex-valued field from intensity measurements - is typically solved using variants of the Gerchberg-Saxton (GS) algorithm, understood as alternating projections between measurement planes. Meanwhile, modern computational imaging increasingly relies on gradient-based optimization and automatic differentiation. Here we show that these two approaches are mathematically identical: the GS magnitude replacement step is exactly a unit gradient descent step on an amplitude least-squares loss. This equivalence enables seamless integration of classical phase retrieval with differentiable physics pipelines. We further identify two complementary probabilistic interpretations of this equivalence: globally, the amplitude loss is the negative log-likelihood under Gaussian amplitude noise; locally, each projection step arises as a Bayesian update with the propagated field as prior. The local view provides qualitative guidance for relaxation in iterative phase retrieval.

Circular Phase Representation and Geometry-Aware Optimization for Ptychographic Image Reconstruction

Traditional iterative reconstruction methods are accurate but computationally expensive, limiting their use in high-throughput and real-time ptychography. Recent deep learning approaches improve speed, but often predict phase as a Euclidean scalar despite its $2π$ periodicity, which can introduce wrapping artifacts, discontinuities at $\pmπ$, and a mismatch between the loss and the underlying signal geometry. We present a deep learning framework for ptychographic reconstruction that models phase on the unit circle using cosine and sine components. Phase error is optimized with a differentiable geodesic loss, which avoids branch-cut discontinuities and provides bounded gradients. The network further incorporates saturation-aware dual-gain input scaling, parallel encoder branches, and three decoders for amplitude, cosine, and sine prediction, together with a composite loss that promotes circular consistency and structural fidelity. Experiments on synthetic and experimental datasets show consistent improvements in both amplitude and phase reconstruction over existing deep learning methods. Frequency-domain analysis further shows better preservation of mid- and high-frequency phase content. The proposed method also provides substantial speedup over iterative solvers while maintaining physically consistent reconstructions.

Adaptable phase retrieval for coherent transition radiation spectroscopy based on differentiable physics information

Coherent transition radiation (CTR) spectroscopy is a critical diagnostic for characterizing the longitudinal structure of relativistic electron bunches in laser-plasma and conventional accelerators. In practice, recovering the bunch profile from a measured CTR spectrum is an ill-posed phase-retrieval problem. Traditionally, this is addressed using Gerchberg-Saxton (GS)-type iterative algorithms. However, these implementations often rely on explicit inverse propagators, making them difficult to adapt to sophisticated experimental forward models. In this work, we introduce a flexible gradient-based framework for CTR phase retrieval. By leveraging a differentiable forward model, we propose a phase-only gradient descent (GD-Phase) approach that enforces the measured spectral amplitude as a hard constraint while optimizing the Fourier phase under physical real-space priors. Using synthetic CTR spectra spanning multi-peaked and strongly modulated profiles, we benchmark GD-Phase against traditional GS and a real-space amplitude-parametrized gradient descent (GD-Amp) algorithm. Unlike traditional methods, this formulation allows for the seamless inclusion of arbitrary differentiable experimental effects into the reconstruction loop. We demonstrate that this physics-informed approach not only reproduces the fidelity of GS methods but also establishes a robust baseline for incorporating multi-diagnostic constraints and uncertainty quantification. This enables the systematic extension to higher-dimensional, multimodal, and uncertainty-aware diagnostics, facilitating fast and scalable phase retrieval in realistic experimental settings.