FB-HyDON: Parameter-Efficient Physics-Informed Operator Learning of Complex PDEs via Hypernetwork and Finite Basis Domain Decomposition

Deep operator networks (DeepONet) and neural operators have gained significant attention for their ability to map infinite-dimensional function spaces and perform zero-shot super-resolution. However, these models often require large datasets for effective training. While physics-informed operators offer a data-agnostic learning approach, they introduce additional training complexities and convergence issues, especially in highly nonlinear systems. To overcome these challenges, we introduce Finite Basis Physics-Informed HyperDeepONet (FB-HyDON), an advanced operator architecture featuring intrinsic domain decomposition. By leveraging hypernetworks and finite basis functions, FB-HyDON effectively mitigates the training limitations associated with existing physics-informed operator learning methods. We validated our approach on the high-frequency harmonic oscillator, Burgers' equation at different viscosity levels, and Allen-Cahn equation demonstrating substantial improvements over other operator learning models.

An Advanced Physics-Informed Neural Operator for Comprehensive Design Optimization of Highly-Nonlinear Systems: An Aerospace Composites Processing Case Study

Deep Operator Networks (DeepONets) and their physics-informed variants have shown significant promise in learning mappings between function spaces of partial differential equations, enhancing the generalization of traditional neural networks. However, for highly nonlinear real-world applications like aerospace composites processing, existing models often fail to capture underlying solutions accurately and are typically limited to single input functions, constraining rapid process design development. This paper introduces an advanced physics-informed DeepONet tailored for such complex systems with multiple input functions. Equipped with architectural enhancements like nonlinear decoders and effective training strategies such as curriculum learning and domain decomposition, the proposed model handles high-dimensional design spaces with significantly improved accuracy, outperforming the vanilla physics-informed DeepONet by two orders of magnitude. Its zero-shot prediction capability across a broad design space makes it a powerful tool for accelerating composites process design and optimization, with potential applications in other engineering fields characterized by strong nonlinearity.

A Sequential Meta-Transfer Learning to Combat Complexities of Physics-Informed Neural Networks: Application to Composites Autoclave Processing

Physics-Informed Neural Networks (PINNs) have gained popularity in solving nonlinear partial differential equations (PDEs) via integrating physical laws into the training of neural networks, making them superior in many scientific and engineering applications. However, conventional PINNs still fall short in accurately approximating the solution of complex systems with strong nonlinearity, especially in long temporal domains. Besides, since PINNs are designed to approximate a specific realization of a given PDE system, they lack the necessary generalizability to efficiently adapt to new system configurations. This entails computationally expensive re-training from scratch for any new change in the system. To address these shortfalls, in this work a novel sequential meta-transfer (SMT) learning framework is proposed, offering a unified solution for both fast training and efficient adaptation of PINNs in highly nonlinear systems with long temporal domains. Specifically, the framework decomposes PDE's time domain into smaller time segments to create "easier" PDE problems for PINNs training. Then for each time interval, a meta-learner is assigned and trained to achieve an optimal initial state for rapid adaptation to a range of related tasks. Transfer learning principles are then leveraged across time intervals to further reduce the computational cost.Through a composites autoclave processing case study, it is shown that SMT is clearly able to enhance the adaptability of PINNs while significantly reducing computational cost, by a factor of 100.