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Abstract Fractional diffusion equations exhibit competitive capabilities in modeling many challenging phenomena such as the anomalously diffusive transport and memory effects. We prove the well‐posedness and regularity of an optimal control of a variably distributed‐order fractional diffusion equation with pointwise constraints, where the distributed‐order operator accounts for, for example, the effect of uncertainties. We accordingly develop and analyze a fully‐discretized finite element approximation to the optimal control without any artificial regularity assumption of the true solution. Numerical experiments are also performed to substantiate the theoretical findings. 

Large Language Models (LLMs) have achieved unprecedented breakthroughs in various natural language processing domains. However, the enigmatic ``black-box'' nature of LLMs remains a significant challenge for interpretability, hampering transparent and accountable applications. While past approaches, such as attention visualization, pivotal subnetwork extraction, and concept-based analyses, offer some insight, they often focus on either local or global explanations within a single dimension, occasionally falling short in providing comprehensive clarity. In response, we propose a novel methodology anchored in sparsity-guided techniques, aiming to provide a holistic interpretation of LLMs. Our framework, termed SparseCBM, innovatively integrates sparsity to elucidate three intertwined layers of interpretation: input, subnetwork, and concept levels. In addition, the newly introduced dimension of interpretable inference-time intervention facilitates dynamic adjustments to the model during deployment. Through rigorous empirical evaluations on real-world datasets, we demonstrate that SparseCBM delivers a profound understanding of LLM behaviors, setting it apart in both interpreting and ameliorating model inaccuracies. Codes are provided in supplements.