Problem

Current Generative Artificial Intelligence (GAI) models, including large language models (LLMs), Generative Adversarial Networks (GANs), and diffusion models, demonstrate impressive capabilities in general domains but face significant limitations in specialized scientific and engineering fields. These limitations stem from data scarcity, the inherent complexity of physical phenomena, and a lack of physical interpretability within their architectures. Traditional methods for data acquisition in these domains, such as experimental testing and high-fidelity simulations, are often prohibitively expensive, time-consuming, or even impossible under extreme conditions, resulting in incomplete datasets. Consequently, applying data-intensive GAI techniques directly to such problems is challenging, as the generated outputs frequently violate fundamental physical laws, rendering them unreliable for critical applications.

Method

The CM-GAI (Continuum Mechanistic Generative Artificial Intelligence) framework introduces a novel theoretical approach that integrates continuum mechanics with optimal transport theory to model the dynamics of data. This method establishes a direct correspondence between concepts in probability space (from optimal transport) and those in continuum mechanics, such as material density, deformation, and kinetic energy, extending the formulation to high-dimensional and non-Euclidean data spaces. A key innovation is the introduction of a "pseudo-time" variable, which can represent various physical parameters like temperature or strain rate. The framework learns the evolution of data probability distributions by minimizing a Lagrangian, subject to a mass conservation constraint that is simplified by working in the reference configuration. External body forces are also incorporated to account for the intrinsic geometry of the data space. Physics-Informed Neural Networks (PINNs) are utilized to solve the governing equations (a second-order partial differential equation for displacement and the mass conservation equation). These networks parameterize the displacement fields and body forces as functions of reference coordinates and pseudo-time. The overall training objective is defined by a loss function that includes terms for matching target probability distributions, enforcing boundary conditions, and satisfying the derived equation of motion, thereby enabling robust data generation from limited samples.

Results

CM-GAI's effectiveness was rigorously demonstrated across three complex engineering problems, showcasing its ability to generate intricate physical data with high accuracy from limited training inputs:

1. Material Level: The framework successfully generated stress-strain responses for a diverse range of materials, including adhesives, glassy polymers, polymer foams, concrete, high-density polyethylene (HDPE), nanotwinned copper (ntCu), glycerol gel, and Polyether Ether Ketone (PEEK). These predictions covered extreme temperatures or strain rates beyond typical experimental ranges. For example, stress-strain curves for an adhesive at 40°C showed good agreement with approximate measurements, and predictions for glassy polymers and polymer foams yielded low Normalized Root Mean Square Errors (NRMSEs) of 1.92% and a reasonable approximation, respectively, demonstrating its capacity to derive constitutive data under challenging conditions.

2. Structure Level: CM-GAI accurately predicted temperature-dependent stress fields within a cantilever beam, even for temperatures outside the training dataset's range. A comparison of the generated stress fields with high-fidelity finite element simulations at an extreme temperature (860°C) revealed a low NRMSE of only 0.56%, effectively demonstrating the method's ability to bypass full thermomechanical coupling simulations at extreme conditions.

3. System Level: The method was successfully applied to generate transient plastic strain fields for a copper Taylor rod impacting a rigid wall at high velocities. The generated plastic equivalent strain (PEEQ) fields exhibited strong agreement with finite element benchmarks, achieving an NRMSE of 1.10% for the 350 m/s impact velocity. This outcome suggests CM-GAI offers an efficient computational pathway for evaluating dynamic responses in mechanical systems, circumventing the need for expensive experiments or complex explicit dynamic finite element simulations prone to convergence issues.

Implications

CM-GAI represents a significant step forward for generative Artificial Intelligence in data-scarce engineering and scientific domains by inherently incorporating physical consistency and offering improved interpretability. By reframing data dynamics within the robust framework of continuum mechanics and leveraging Physics-Informed Neural Networks, the approach can accurately extrapolate complex material and structural behaviors under extreme conditions with only limited training data. This capability substantially reduces the reliance on costly experiments and computationally intensive simulations. The framework's innovative consideration of deformation energy in feature space and its generalization of optimal transport provide a powerful theoretical foundation for developing future "world models" that intrinsically understand physical laws. Future research will focus on extending the theory and numerical methods to even higher-dimensional data, formulating more sophisticated constitutive laws for feature space deformation, and integrating advanced manifold learning techniques to fully capture the intrinsic geometry of underlying data structures.