Monotone Peridynamic Neural Operator for Nonlinear Material Modeling with Conditionally Unique Solutions.

Area of Science:

• Computational mechanics
• Materials science
• Machine learning

Background:

• Nonlocal continuum mechanics models like peridynamics are crucial for simulating complex materials.
• Current data-driven methods for peridynamics lack guaranteed well-posedness, leading to potential non-physical results.
• Accurate and reliable data-driven models are needed to streamline material characterization.

Purpose of the Study:

• To develop a data-driven nonlocal constitutive model with guaranteed well-posedness and solution uniqueness.
• To introduce the monotone peridynamic neural operator (MPNO) as a solution.
• To validate MPNO's performance on synthetic and real-world data.

Main Methods:

• MPNO learns a nonlocal kernel and constitutive relation using a monotone gradient network.
• This architectural constraint ensures the convexity of the learned energy density function.
• Guaranteed uniqueness of solutions in the small deformation regime is achieved.

Main Results:

• MPNO converges to the ground-truth model on synthetic data as measurement grid size decreases.
• MPNO demonstrates superior generalization compared to conventional neural networks on unseen data.
• MPNO successfully learns a homogenized model from molecular dynamics data, showing practical utility.

Conclusions:

• MPNO provides a robust framework for data-driven nonlocal material modeling with guaranteed well-posedness.
• The approach enhances accuracy and generalization in peridynamic simulations.
• MPNO offers a physically interpretable and expressive tool for materials science applications.