Ning Liu

Neural operators (NOs) have demonstrated remarkable success in learning mappings between function spaces, serving as efficient approximators for the forward solutions of complex physical systems governed by partial differential equations (PDEs). However, while effective as black-box solvers, they offer limited insight into the underlying physical mechanism, due to the lack of interpretable representations of the physical parameters that drive the system. To tackle this challenge, we propose a new paradigm for learning disentangled representations from neural operator parameters, thereby effectively solving an inverse problem. Specifically, we introduce DisentangO, a novel hyper-neural operator architecture designed to unveil and disentangle the latent physical factors of variation embedded within the black-box neural operator parameters. At the core of DisentangO is a multi-task neural operator architecture that distills the varying parameters of the governing PDE through a task-wise adaptive layer, coupled with a hierarchical variational autoencoder that disentangles these variations into identifiable latent factors. By learning these disentangled representations, our model not only enhances physical interpretability but also enables more robust generalization across diverse physical systems. Empirical evaluations across supervised, semi-supervised, and unsupervised learning contexts show that DisentangO effectively extracts meaningful and interpretable latent features, bridging the divide between predictive performance and physical understanding in neural operator frameworks.

Despite the recent popularity of attention-based neural architectures in core AI fields like natural language processing (NLP) and computer vision (CV), their potential in modeling complex physical systems remains under-explored. Learning problems in physical systems are often characterized as discovering operators that map between function spaces based on a few instances of function pairs. This task frequently presents a severely ill-posed PDE inverse problem. In this work, we propose a novel neural operator architecture based on the attention mechanism, which we coin Nonlocal Attention Operator (NAO), and explore its capability towards developing a foundation physical model. In particular, we show that the attention mechanism is equivalent to a double integral operator that enables nonlocal interactions among spatial tokens, with a data-dependent kernel characterizing the inverse mapping from data to the hidden parameter field of the underlying operator. As such, the attention mechanism extracts global prior information from training data generated by multiple systems, and suggests the exploratory space in the form of a nonlinear kernel map. Consequently, NAO can address ill-posedness and rank deficiency in inverse PDE problems by encoding regularization and achieving generalizability. We empirically demonstrate the advantages of NAO over baseline neural models in terms of generalizability to unseen data resolutions and system states. Our work not only suggests a novel neural operator architecture for learning interpretable foundation models of physical systems, but also offers a new perspective towards understanding the attention mechanism.

Large language models (LLMs) bear promise as a fast and accurate material modeling paradigm for evaluation, analysis, and design. Their vast number of trainable parameters necessitates a wealth of data to achieve accuracy and mitigate overfitting. However, experimental measurements are often limited and costly to obtain in sufficient quantities for finetuning. To this end, we present a physics-based training pipeline that tackles the pathology of data scarcity. The core enabler is a physics-based modeling framework that generates a multitude of synthetic data to align the LLM to a physically consistent initial state before finetuning. Our framework features a two-phase training strategy: (1) utilizing the large-in-amount while less accurate synthetic data for supervised pretraining, and (2) finetuning the phase-1 model with limited experimental data. We empirically demonstrate that supervised pretraining is vital to obtaining accurate finetuned LLMs, via the lens of learning polymer flammability metrics where cone calorimeter data is sparse.

Neural operators (NOs) have emerged as effective tools for modeling complex physical systems in scientific machine learning. In NOs, a central characteristic is to learn the governing physical laws directly from data. In contrast to other machine learning applications, partial knowledge is often known a priori about the physical system at hand whereby quantities such as mass, energy and momentum are exactly conserved. Currently, NOs have to learn these conservation laws from data and can only approximately satisfy them due to finite training data and random noise. In this work, we introduce conservation law-encoded neural operators (clawNOs), a suite of NOs that endow inference with automatic satisfaction of such conservation laws. ClawNOs are built with a divergence-free prediction of the solution field, with which the continuity equation is automatically guaranteed. As a consequence, clawNOs are compliant with the most fundamental and ubiquitous conservation laws essential for correct physical consistency. As demonstrations, we consider a wide variety of scientific applications ranging from constitutive modeling of material deformation, incompressible fluid dynamics, to atmospheric simulation. ClawNOs significantly outperform the state-of-the-art NOs in learning efficacy, especially in small-data regimes.

Fourier neural operators (FNOs) can learn highly nonlinear mappings between function spaces, and have recently become a popular tool for learning responses of complex physical systems. However, to achieve good accuracy and efficiency, FNOs rely on the Fast Fourier transform (FFT), which is restricted to modeling problems on rectangular domains. To lift such a restriction and permit FFT on irregular geometries as well as topology changes, we introduce domain agnostic Fourier neural operator (DAFNO), a novel neural operator architecture for learning surrogates with irregular geometries and evolving domains. The key idea is to incorporate a smoothed characteristic function in the integral layer architecture of FNOs, and leverage FFT to achieve rapid computations, in such a way that the geometric information is explicitly encoded in the architecture. In our empirical evaluation, DAFNO has achieved state-of-the-art accuracy as compared to baseline neural operator models on two benchmark datasets of material modeling and airfoil simulation. To further demonstrate the capability and generalizability of DAFNO in handling complex domains with topology changes, we consider a brittle material fracture evolution problem. With only one training crack simulation sample, DAFNO has achieved generalizability to unseen loading scenarios and substantially different crack patterns from the trained scenario. Our code and data accompanying this paper are available at https://github.com/ningliu-iga/DAFNO.

Neural operators, which emerge as implicit solution operators of hidden governing equations, have recently become popular tools for learning responses of complex real-world physical systems. Nevertheless, the majority of neural operator applications has thus far been data-driven, which neglects the intrinsic preservation of fundamental physical laws in data. In this paper, we introduce a novel integral neural operator architecture, to learn physical models with fundamental conservation laws automatically guaranteed. In particular, by replacing the frame-dependent position information with its invariant counterpart in the kernel space, the proposed neural operator is designed to be translation- and rotation-invariant, and consequently abides by the conservation laws of linear and angular momentums. As applications, we demonstrate the expressivity and efficacy of our model in learning complex material behaviors from both synthetic and experimental datasets, and show that, by automatically satisfying these essential physical laws, our learned neural operator is not only generalizable in handling translated and rotated datasets, but also achieves improved accuracy and efficiency from the baseline neural operator models.