Qianying Cao

Qianying Cao, Somdatta Goswami, George Em Karniadakis

We introduce the Laplace neural operator (LNO), which leverages the Laplace transform to decompose the input space. Unlike the Fourier Neural Operator (FNO), LNO can handle non-periodic signals, account for transient responses, and exhibit exponential convergence. LNO incorporates the pole-residue relationship between the input and the output space, enabling greater interpretability and improved generalization ability. Herein, we demonstrate the superior approximation accuracy of a single Laplace layer in LNO over four Fourier modules in FNO in approximating the solutions of three ODEs (Duffing oscillator, driven gravity pendulum, and Lorenz system) and three PDEs (Euler-Bernoulli beam, diffusion equation, and reaction-diffusion system). Notably, LNO outperforms FNO in capturing transient responses in undamped scenarios. For the linear Euler-Bernoulli beam and diffusion equation, LNO's exact representation of the pole-residue formulation yields significantly better results than FNO. For the nonlinear reaction-diffusion system, LNO's errors are smaller than those of FNO, demonstrating the effectiveness of using system poles and residues as network parameters for operator learning. Overall, our results suggest that LNO represents a promising new approach for learning neural operators that map functions between infinite-dimensional spaces.

Heechang Kim, Qianying Cao, Hyomin Shin et al.

Neural operators have emerged as fast surrogate solvers for parametric partial differential equations (PDEs). However, purely data-driven models often require extensive training data and can generalize poorly, especially in small-data regimes and under unseen (out-of-distribution) input functions that are not represented in the training data. To address these limitations, we propose the Physics-Informed Laplace Neural Operator (PILNO), which enhances the Laplace Neural Operator (LNO) by embedding governing physics into training through PDE, boundary condition, and initial condition residuals. To improve expressivity, we first introduce an Advanced LNO (ALNO) backbone that retains a pole-residue transient representation while replacing the steady-state branch with an FNO-style Fourier multiplier. To make physics-informed training both data-efficient and robust, PILNO further leverages (i) virtual inputs: an unlabeled ensemble of input functions spanning a broad spectral range that provides abundant physics-only supervision and explicitly targets out-of-distribution (OOD) regimes; and (ii) temporal-causality weighting: a time-decaying reweighting of the physics residual that prioritizes early-time dynamics and stabilizes optimization for time-dependent PDEs. Across four representative benchmarks -- Burgers' equation, Darcy flow, a reaction-diffusion system, and a forced KdV equation -- PILNO consistently improves accuracy in small-data settings (e.g., N_train <= 27), reduces run-to-run variability across random seeds, and achieves stronger OOD generalization than purely data-driven baselines.

Qianying Cao, Shanqing Liu, Alan John Varghese et al.

Time series forecasting plays a pivotal role in a wide range of applications, including weather prediction, healthcare, structural health monitoring, predictive maintenance, energy systems, and financial markets. While models such as LSTM, GRU, Transformers, and State-Space Models (SSMs) have become standard tools in this domain, selecting the optimal architecture remains a challenge. Performance comparisons often depend on evaluation metrics and the datasets under analysis, making the choice of a universally optimal model controversial. In this work, we introduce a flexible automated framework for time series forecasting that systematically designs and evaluates diverse network architectures by integrating LSTM, GRU, multi-head Attention, and SSM blocks. Using a multi-objective optimization approach, our framework determines the number, sequence, and combination of blocks to align with specific requirements and evaluation objectives. From the resulting Pareto-optimal architectures, the best model for a given context is selected via a user-defined preference function. We validate our framework across four distinct real-world applications. Results show that a single-layer GRU or LSTM is usually optimal when minimizing training time alone. However, when maximizing accuracy or balancing multiple objectives, the best architectures are often composite designs incorporating multiple block types in specific configurations. By employing a weighted preference function, users can resolve trade-offs between objectives, revealing novel, context-specific optimal architectures. Our findings underscore that no single neural architecture is universally optimal for time series forecasting. Instead, the best-performing model emerges as a data-driven composite architecture tailored to user-defined criteria and evaluation objectives.

Nazanin Ahmadi, Qianying Cao, Jay D. Humphrey et al.

Physics-informed machine learning (PIML) is emerging as a potentially transformative paradigm for modeling complex biomedical systems by integrating parameterized physical laws with data-driven methods. Here, we review three main classes of PIML frameworks: physics-informed neural networks (PINNs), neural ordinary differential equations (NODEs), and neural operators (NOs), highlighting their growing role in biomedical science and engineering. We begin with PINNs, which embed governing equations into deep learning models and have been successfully applied to biosolid and biofluid mechanics, mechanobiology, and medical imaging among other areas. We then review NODEs, which offer continuous-time modeling, especially suited to dynamic physiological systems, pharmacokinetics, and cell signaling. Finally, we discuss deep NOs as powerful tools for learning mappings between function spaces, enabling efficient simulations across multiscale and spatially heterogeneous biological domains. Throughout, we emphasize applications where physical interpretability, data scarcity, or system complexity make conventional black-box learning insufficient. We conclude by identifying open challenges and future directions for advancing PIML in biomedical science and engineering, including issues of uncertainty quantification, generalization, and integration of PIML and large language models.

David S. Li, Somdatta Goswami, Qianying Cao et al.

Thoracic aortic aneurysms (TAAs) arise from diverse mechanical and mechanobiological disruptions to the aortic wall that increase the risk of dissection or rupture. Evidence links TAA development to dysfunctions in the aortic mechanotransduction axis, including loss of elastic fiber integrity and cell-matrix connections. Because distinct insults create different mechanical vulnerabilities, there is a critical need to identify interacting factors that drive progression. Here, we use a finite element framework to generate synthetic TAAs from hundreds of heterogeneous insults spanning varying degrees of elastic fiber damage and impaired mechanosensing. From these simulations, we construct spatial maps of localized dilatation and distensibility to train neural networks that predict the initiating combined insult. We compare several architectures (Deep Operator Networks, UNets, and Laplace Neural Operators) and multiple input data formats to define a standard for future subject-specific modeling. We also quantify predictive performance when networks are trained using only geometric data (dilatation) versus both geometric and mechanical data (dilatation plus distensibility). Across all networks, prediction errors are significantly higher when trained on dilatation alone, underscoring the added value of distensibility information. Among the tested models, UNet consistently provides the highest accuracy across all data formats. These findings highlight the importance of acquiring full-field measurements of both dilatation and distensibility in TAA assessment to reveal the mechanobiological drivers of disease and support the development of personalized treatment strategies.