Liquid Fourier Latent Dynamics Networks for fast GPU-based numerical simulations in computational cardiology

TL;DR

This work introduces Liquid Fourier Latent Dynamics Networks (LFLDNets), an extension of Latent Dynamics Networks designed to learn parameterized space-time surrogates for nonlinear multiscale PDEs on complex geometries. By coupling a CfC,NCP-based Liquid Neural Network for temporal evolution with a Fourier-encoded reconstruction network, LFLDNets achieve accurate, low-parameter surrogates that run efficiently on GPUs. The authors demonstrate two 3D, patient-specific cardiovascular test cases—cardiac electrophysiology and CFD in the aorta—showing fast inference (minutes) with competitive accuracy compared to high-fidelity simulations and improved generalization due to the Fourier embedding and sparse dynamics. This framework advances digital twin capabilities by enabling rapid, parameterized space-time simulations across challenging multiphysics scenarios.

Abstract

Scientific Machine Learning (ML) is gaining momentum as a cost-effective alternative to physics-based numerical solvers in many engineering applications. In fact, scientific ML is currently being used to build accurate and efficient surrogate models starting from high-fidelity numerical simulations, effectively encoding the parameterized temporal dynamics underlying Ordinary Differential Equations (ODEs), or even the spatio-temporal behavior underlying Partial Differential Equations (PDEs), in appropriately designed neural networks. We propose an extension of Latent Dynamics Networks (LDNets), namely Liquid Fourier LDNets (LFLDNets), to create parameterized space-time surrogate models for multiscale and multiphysics sets of highly nonlinear differential equations on complex geometries. LFLDNets employ a neurologically-inspired, sparse, liquid neural network for temporal dynamics, relaxing the requirement of a numerical solver for time advancement and leading to superior performance in terms of tunable parameters, accuracy, efficiency and learned trajectories with respect to neural ODEs based on feedforward fully-connected neural networks. Furthermore, in our implementation of LFLDNets, we use a Fourier embedding with a tunable kernel in the reconstruction network to learn high-frequency functions better and faster than using space coordinates directly as input. We challenge LFLDNets in the framework of computational cardiology and evaluate their capabilities on two 3-dimensional test cases arising from multiscale cardiac electrophysiology and cardiovascular hemodynamics. This paper illustrates the capability to run Artificial Intelligence-based numerical simulations on single or multiple GPUs in a matter of minutes and represents a significant step forward in the development of physics-informed digital twins.

Figures (8)

• Figure 1: Comparison between Latent Dynamics Networks (left) and Liquid Fourier Latent Dynamics Networks (right).
• Figure 2: Electrophysiology test case. Comparison of the action potential $u(\boldsymbol{x}, t)$ throughout the heartbeat for a random sample in the validation set. LFLDNet prediction (left), ground truth from electrophysiology simulation (right).
• Figure 3: Electrophysiology test case. Pointwise absolute difference $|u_\mathrm{pred}(\boldsymbol{x}, t) - u_\mathrm{obs}(\boldsymbol{x}, t)|$ between LFLDNets predictions and observations over the cardiac cycle for a random sample in the validation set.
• Figure 4: Electrophysiology test case. Comparison of training and validation losses between LDNets, LLDNets, and LFLDNets after hyperparameter tuning (top). Comparison of training and validation losses for different Fourier embedding dimensions within the optimal LFLDNets (bottom).
• Figure 5: Electrophysiology test case. Evolution of the global state vector $\boldsymbol{s}(t)$ over the cardiac cycle for a specific choice of the input signals $\boldsymbol{I}(t)$ coming from a random sample in the validation set. Different colors identify different components of $\boldsymbol{s}(t)$. Optimal LDNet (top), optimal LLDNet (center) and optimal LFLDNet (bottom).