Namjung Kim

Junho Choi, Taehyun Yun, Namjung Kim et al.

In this paper, we introduce the Spectral Coefficient Learning via Operator Network (SCLON), a novel operator learning-based approach for solving parametric partial differential equations (PDEs) without the need for data harnessing. The cornerstone of our method is the spectral methodology that employs expansions using orthogonal functions, such as Fourier series and Legendre polynomials, enabling accurate PDE solutions with fewer grid points. By merging the merits of spectral methods - encompassing high accuracy, efficiency, generalization, and the exact fulfillment of boundary conditions - with the prowess of deep neural networks, SCLON offers a transformative strategy. Our approach not only eliminates the need for paired input-output training data, which typically requires extensive numerical computations, but also effectively learns and predicts solutions of complex parametric PDEs, ranging from singularly perturbed convection-diffusion equations to the Navier-Stokes equations. The proposed framework demonstrates superior performance compared to existing scientific machine learning techniques, offering solutions for multiple instances of parametric PDEs without harnessing data. The mathematical framework is robust and reliable, with a well-developed loss function derived from the weak formulation, ensuring accurate approximation of solutions while exactly satisfying boundary conditions. The method's efficacy is further illustrated through its ability to accurately predict intricate natural behaviors like the Kolmogorov flow and boundary layers. In essence, our work pioneers a compelling avenue for parametric PDE solutions, serving as a bridge between traditional numerical methodologies and cutting-edge machine learning techniques in the realm of scientific computation.

Junho Choi, Namjung Kim, Youngjoon Hong

Machine learning methods have been lately used to solve partial differential equations (PDEs) and dynamical systems. These approaches have been developed into a novel research field known as scientific machine learning in which techniques such as deep neural networks and statistical learning are applied to classical problems of applied mathematics. In this paper, we develop a novel numerical algorithm that incorporates machine learning and artificial intelligence to solve PDEs. Based on the Legendre-Galerkin framework, we propose the {\it unsupervised machine learning} algorithm to learn {\it multiple instances} of the solutions for different types of PDEs. Our approach overcomes the limitations of data-driven and physics-based methods. The proposed neural network is applied to general 1D and 2D PDEs with various boundary conditions as well as convection-dominated {\it singularly perturbed PDEs} that exhibit strong boundary layer behavior.

Yeongbin Cha, Namjung Kim

Additive manufacturing (AM) relies critically on understanding and extrapolating process-property relationships; however, existing data-driven approaches remain limited by fragmented knowledge representations and unreliable extrapolation under sparse data conditions. In this study, we propose an ontology-guided, equation-centric framework that tightly integrates large language models (LLMs) with an additive manufacturing mathematical knowledge graph (AM-MKG) to enable reliable knowledge extraction and principled extrapolative modeling. By explicitly encoding equations, variables, assumptions, and their semantic relationships within a formal ontology, unstructured literature is transformed into machine-interpretable representations that support structured querying and reasoning. LLM-based equation generation is further conditioned on MKG-derived subgraphs, enforcing physically meaningful functional forms and mitigating non-physical or unstable extrapolation trends. To assess reliability beyond conventional predictive uncertainty, a confidence-aware extrapolation assessment is introduced, integrating extrapolation distance, statistical stability, and knowledge-graph-based physical consistency into a unified confidence score. Results demonstrate that ontology-guided extraction significantly improves the structural coherence and quantitative reliability of extracted knowledge, while subgraph-conditioned equation generation yields stable and physically consistent extrapolations compared to unguided LLM outputs. Overall, this work establishes a unified pipeline for ontology-driven knowledge representation, equation-centered reasoning, and confidence-based extrapolation assessment, highlighting the potential of knowledge-graph-augmented LLMs as reliable tools for extrapolative modeling in additive manufacturing.

Junho Choi, Teng-Yuan Chang, Namjung Kim et al.

Ensemble simulations of high-dimensional flow models (e.g., Navier Stokes type PDEs) are computationally prohibitive for real time applications. Neural operators enable fast inference but are limited by costly data requirements and poor generalization to 3D flows. We present a data-free operator network for the Navier Stokes equations that eliminates the need for paired solution data and enables robust, real time inference for large ensemble forecasting. The physics-grounded architecture takes initial and boundary conditions as well as forcing functions, yielding solutions robust to high variability and perturbations. Across 2D benchmarks and 3D test cases, the method surpasses prior neural operators in accuracy and, for ensembles, achieves greater efficiency than conventional numerical solvers. Notably, it delivers accurate solutions of the three dimensional Navier Stokes equations, a regime not previously demonstrated for data free neural operators. By uniting a numerically grounded architecture with the scalability of machine learning, this approach establishes a practical pathway toward data free, high fidelity PDE surrogates for end to end scientific simulation and prediction.

Namjung Kim, Dongseok Lee, Jongbin Yu et al.

Advances in material functionalities drive innovations across various fields, where metamaterials-defined by structure rather than composition-are leading the way. Despite the rise of artificial intelligence (AI)-driven design strategies, their impact is limited by task-specific retraining, poor out-of-distribution(OOD) generalization, and the need for separate models for forward and inverse design. To address these limitations, we introduce the Metamaterial Foundation Model (MetaFO), a Bayesian transformer-based foundation model inspired by large language models. MetaFO learns the underlying mechanics of metamaterials, enabling probabilistic, zero-shot predictions across diverse, unseen combinations of material properties and structural responses. It also excels in nonlinear inverse design, even under OOD conditions. By treating metamaterials as an operator that maps material properties to structural responses, MetaFO uncovers intricate structure-property relationships and significantly expands the design space. This scalable and generalizable framework marks a paradigm shift in AI-driven metamaterial discovery, paving the way for next-generation innovations.