Schr\"odinger-Inspired Time-Evolution for 4D Deformation Forecasting

TL;DR

This paper introduces a Schr"odinger-inspired neural network architecture for stable, interpretable, and physics-guided 4D (3D + time) deformation forecasting, applicable to medical imaging and physical phenomena.

Contribution

It presents the first end-to-end 4D forecasting model embedding a Schr"odinger-type evolution operator within a deep learning framework, enhancing stability and interpretability.

Findings

Achieves accurate, stable 4D predictions on synthetic benchmarks.

Provides interpretable latent representations encoding transport and interactions.

Demonstrates robustness and anatomical fidelity in medical imaging simulations.

Abstract

Spatiotemporal forecasting of complex three-dimensional phenomena (4D: 3D + time) is fundamental to applications in medical imaging, fluid and material dynamics, and geophysics. In contrast to unconstrained neural forecasting models, we propose a Schr\"odinger-inspired, physics-guided neural architecture that embeds an explicit time-evolution operator within a deep convolutional framework for 4D prediction. From observed volumetric sequences, the model learns voxelwise amplitude, phase, and potential fields that define a complex-valued wavefunction $\psi = A e^{i\phi}$, which is evolved forward in time using a differentiable, unrolled Schr\"odinger time stepper. This physics-guided formulation yields several key advantages: (i) temporal stability arising from the structured evolution operator, which mitigates drift and error accumulation in long-horizon forecasting; (ii) an interpretable latent representation, where phase encodes transport dynamics, amplitude captures structural intensity, and the learned potential governs spatiotemporal interactions; and (iii) natural compatibility with deformation-based synthesis, which is critical for preserving anatomical fidelity in medical imaging applications. By integrating physical priors directly into the learning process, the proposed approach combines the expressivity of deep networks with the robustness and interpretability of physics-based modeling. We demonstrate accurate and stable prediction of future 4D states, including volumetric intensities and deformation fields, on synthetic benchmarks that emulate realistic shape deformations and topological changes. To our knowledge, this is the first end-to-end 4D neural forecasting framework to incorporate a Schr\"odinger-type evolution operator, offering a principled pathway toward interpretable, stable, and anatomically consistent spatiotemporal prediction.