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      "statement": "# DeepXDE neural differential-equation software with adaptive residual refinement\n\nImplements PINN problem specification, geometry/boundary handling and training workflows, with residual-based adaptive refinement that adds training points at high sampled residuals.\n\n## Scientific scope\n\nFour original authors; stars are institutional affiliations, not equal-contribution marks. Software/RAR contribution distinct from original PINN method. Inherits derivative approximation theory; sampling/optimization and quadrature errors remain. RAR Monte Carlo stopping is not a rigorous global error certificate or unrestricted PDE convergence.\n\n## Reading status\n\nThis is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.\n\n## People and systems\n\nLu Lu; Xuhui Meng; Zhiping Mao; George E. Karniadakis\n\nSource-listed authors or explicitly credited technical contributors for the cited versions. Family entries retain source-specific attribution; a combined list is not one original-paper byline or equal individual work. Names alone do not establish resolved identities.\n\n\n\n## Original sources\n\n- [DeepXDE: A deep learning library for solving differential equations](https://arxiv.org/pdf/1907.04502v1)\n",
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