The Conforming Decomposition Finite Element Method (CDFEM) [1] and the Conforming Transient h-r Unstructured Adaptive Mesh Refinement (cThruAMR) method were developed to generate interface-conforming finite element meshes for coupled multiphase and multimaterial problems with moving interfaces. In CDFEM, level sets are used to define the domains of individual materials or phases. Nodes are introduced at the intersections of level set surfaces with mesh edges, enabling automatic generation of conforming meshes that accurately capture both weak and strong discontinuities across interfaces using standard finite element methods. This approach has been successfully applied to dynamic interface problems, including laser welding, metal additive manufacturing, direct writing of ceramics, and the ablation of thermal protection systems. cThruAMR combines interface-conforming h-adaptivity (mesh refinement or cutting) and r-adaptivity (node repositioning) to produce high-quality meshes that conform to evolving interfaces. While h-adaptivity effectively captures dynamic interface geometry, it can result in a proliferation of infinitesimal finite elements. To address this, r-adaptivity improves mesh quality by repositioning nodes to capture desired features with significantly fewer cut elements. By integrating h- and r-adaptivity, cThruAMR achieves robust mesh generation for complex, evolving geometries. This work extends CDFEM and cThruAMR to utilize a novel geometry description: an Implicit Neural Representation (INR) of a Signed Distance Function (SDF). The objective is to generate high-quality interface-conforming meshes for INR-based SDFs, bridging the gap between neural modeling and computational applications. Adaptive strategies are employed to refine the background mesh, followed by interface-conforming h- and r-adaptivity to accurately capture the implicit interface encoded in the INR. Verification simulations demonstrate that the proposed method effectively captures geometry with conforming meshes and enables convergent transport simulations using the resulting discretizations. *Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.