Design-GenNO: A physics-informed generative model with neural operators for inverse microstructure design

• We introduce Design-GenNO, a physics-informed probabilistic framework that integrates deep generative modeling, neural operators, and PDE-constrained training for inverse microstructure design. • Latent variables encode both microstructures and full PDE solution fields into a structured, low-dimensional space, enabling efficient exploration of the design space. Training leverages virtual PDE residuals, reducing reliance on costly labeled solution data. • By capturing the entire PDE solution field, Design-GenNO allows solving a wide variety of design problems for different target properties without retraining. A learnable normalizing flow prior captures complex, potentially multimodal latent distributions, improving sampling efficiency and convergence. • Design-GenNO produces accurate and diverse microstructures and solution fields, effectively satisfies various design objectives, outperforms existing methods in predictive fidelity, and demonstrates robust extrapolation beyond the training dataset. Inverse microstructure design plays a central role in materials discovery, yet remains challenging due to the complexity of structure–property linkages and the scarcity of labeled training data. We propose Design-GenNO, a physics-informed generative neural operator framework that unifies generative modeling with operator learning to address these challenges. In Design-GenNO, microstructures are encoded into a low-dimensional, well-structured latent space, which serves as the generator for both reconstructing microstructures and predicting solution fields of governing PDEs. MultiONet-based decoders enable functional mappings from latent variables to both microstructures and full PDE solution fields, allowing a multitude of design objectives to be addressed without retraining. A normalizing flow prior regularizes the latent space, facilitating efficient sampling and robust gradient-based optimization. A distinctive feature of the framework is its physics-informed training strategy: by embedding PDE residuals directly into the learning objective, Design-GenNO significantly reduces reliance on labeled datasets and can even operate in a self-supervised setting. We validate the method on a suite of inverse design tasks in two-phase materials, including effective property matching, recovery of microstructures from sparse field measurements, and maximization of conductivity ratios. Across all tasks, Design-GenNO achieves high accuracy, generates diverse and physically meaningful designs, and consistently outperforms the state-of-the-art method. Moreover, it demonstrates strong extrapolative capabilities by producing microstructures with effective properties beyond those in the training data. These results establish Design-GenNO as a robust and general framework for physics-informed inverse design, offering a promising pathway toward accelerated materials discovery.