Neural implicit representations have emerged as a powerful paradigm for representing 3D geometry. However, evolving these representations under physical dynamics or user-specified deformations remains challenging, as the implicit function must be updated globally even for local changes. We present a level set theory that enables neural implicit surfaces to evolve under explicit flow fields. Our key insight is to derive a partial differential equation (PDE) governing the evolution of the neural network weights, such that the zero level set of the network tracks the evolving surface exactly. This formulation allows us to deform neural implicit surfaces using classical flow fields (mean curvature flow, advection) while maintaining the advantages of implicit representations.

The level set method, pioneered by Osher and Sethian, represents surfaces as the zero level set of a higher-dimensional function and evolves this function according to a PDE derived from the desired surface motion. This approach has been enormously successful in computational physics, fluid dynamics, and classical computer vision. Recently, neural implicit functions (e.g., DeepSDF, NeRF) have demonstrated remarkable ability to represent complex 3D geometry. However, these representations are typically static — once trained, the surface they encode cannot be easily modified without retraining. We ask: can we derive a principled framework for evolving neural implicit surfaces, analogous to the classical level set method?

The fundamental challenge is that in the classical level set method, the level set function is discretized on a grid and each grid point can be updated independently. In contrast, a neural implicit function is parameterized by network weights that jointly determine the function value everywhere — changing one weight affects the entire surface. Our approach resolves this by deriving the chain rule relationship between the level set PDE and the network weight dynamics, showing that the weight update rule can be expressed as a pseudo-inverse problem involving the Jacobian of the network.

Let f_theta(x): R^3 -> R be a neural implicit function parameterized by weights theta, where the surface S is defined as S = {x : f_theta(x) = 0}. Given a velocity field v(x, t) that specifies how each point on the surface should move, the classical level set equation states df/dt + v . grad(f) = 0. For a neural network, we have df/dt = (df/d_theta) . (d_theta/dt), where df/d_theta is the Jacobian of the network output with respect to its weights. Substituting and evaluating on the surface, we obtain a linear system J(d_theta/dt) = -v . grad_x(f) that can be solved for the weight update d_theta/dt using least squares.

In practice, we sample points on the current zero level set and compute the Jacobian at these points. The resulting overdetermined linear system is solved using damped least squares (Tikhonov regularization) to ensure stability. We demonstrate the framework with several classical flows: mean curvature flow (which smooths surfaces), advection by a velocity field (which translates and deforms surfaces), and normal flow (which inflates or deflates surfaces). The method naturally handles topological changes such as merging and splitting, inheriting this capability from the level set formulation.

We validate our approach on several tasks. First, we demonstrate mean curvature flow on neural implicit surfaces represented by SIREN networks. Starting from a noisy surface, our method smoothly evolves the surface toward a sphere (the minimal surface), correctly reproducing the known analytical behavior of curvature flow. Compared to naive approaches that retrain the network at each timestep, our method is 10-50x faster while producing more accurate surface evolution, as the weight update directly encodes the desired motion rather than relying on fitting to a moved point cloud.

Second, we apply our framework to shape deformation by user-specified velocity fields. Given a neural implicit shape (e.g., a bunny or dragon mesh encoded as a neural SDF), users can specify local deformations and our method propagates the deformation smoothly through the weight update, maintaining surface quality without artifacts. Third, we demonstrate topology-changing flows where two initially separate objects merge into one as they are advected toward each other. The neural implicit representation seamlessly handles the topological transition, which would require explicit remeshing in traditional approaches.

We have presented a principled level set theory for evolving neural implicit surfaces under explicit flow fields. By deriving the relationship between the surface evolution PDE and the network weight dynamics, we enable efficient, accurate, and topology-aware deformation of neural implicit representations. Our framework bridges classical computational geometry and modern neural scene representations, opening new possibilities for physics-based animation, shape optimization, and interactive geometry processing with neural implicit functions.