ViscoReg: Viscosity Regularization Overview

• ViscoReg is a framework integrating viscous penalization to stabilize ill-posed PDE problems across neural, imaging, and simulation domains.
• It improves neural SDF training by incorporating a decaying Laplacian term that enhances geometric reconstruction metrics like Chamfer and Hausdorff distances.
• The approach extends to viscoelastic modulus recovery, rate-independent evolution, robotic manipulation, and high-Weissenberg fluid simulations, validated by rigorous analysis and experiments.

ViscoReg encompasses a set of methodologies and frameworks leveraging viscosity-based regularization principles to enhance stability, accuracy, and physical fidelity in problems involving partial differential equations (PDEs), inference of material properties, and time-incremental evolution for complex systems. The hallmark of ViscoReg approaches is the use of viscous penalization or viscous regularization—often based on vanishing-viscosity theory, adjoint methods, or conformation-tensor changes of variables—to ensure stable, physically meaningful solutions across neural, computational, and continuum-mechanics domains.

1. Viscosity Regularization in Neural Signed Distance Function Training

ViscoReg advances the training of Neural Signed Distance Functions (SDFs) for 3D scene reconstruction by augmenting standard Eikonal-based losses with viscous regularization techniques that enforce uniqueness and stability. The central objective functions are:

• Eikonal equation enforcement: $\,\|\nabla \phi(x)\|_2 = 1$, $\phi|_{\partial\Omega}=0$ for the SDF.
• Loss function:

$$L_{Eikonal}(\theta) = \mathbb{E}_{x\sim\mu}\left[(\|\nabla f_\theta(x)\|_2 -1)^2\right]$$

with a manifold fitting term $L_m(\theta)$.

However, the Eikonal equation is ill-posed, admitting infinitely many weak (non-distance) solutions. Gradient descent on the Eikonal loss yields a forward–backward heat equation exhibiting unstable dynamics and high-frequency blowup.

ViscoReg replaces the residual with a viscous term:

$$r_\varepsilon(u) = \|\nabla u(x)\|_2 -1 - \varepsilon\,\Delta u(x)$$

with annealing of $\varepsilon \to 0$ during training. The ViscoReg loss thus incorporates a decaying Laplacian penalization, resulting in a stabilized fourth-order PDE for the gradient flow. High-frequency modes are damped, and the system converges to the unique viscosity solution—the actual signed distance function. Empirical benchmarks on surface and scene reconstruction (Chamfer and Hausdorff distances, IoU) show ViscoReg consistently outperforming state-of-the-art INR methods such as SIREN, DiGS, and StEik, with negligible added computational cost (Krishnan et al., 1 Jul 2025).

2. Viscous Regularization for Viscoelastic Modulus Reconstruction

In the domain of viscoelastic imaging, ViscoReg refers to an iterative adjoint-based gradient descent method for reconstructing spatially-varying shear modulus $\mu(x)$ and viscosity $\eta_\mu(x)$ from internal displacement measurements under time-harmonic excitation (Ammari et al., 2014). Governed by the nearly incompressible Stokes system:

$$\nabla \cdot [ (\mu + i\omega\eta_\mu)\nabla^s u ] + \nabla p + \rho\omega^2 u = 0, \quad \nabla\cdot u = 0$$

the inverse problem is formulated as minimizing the least-squares discrepancy between measured and modeled displacements:

$$J(\mu,\eta_\mu) = \frac12\int_\Omega |u(\mu,\eta_\mu) - u_{meas}|^2\,dx$$

Fréchet derivatives of $J$—evaluated via an adjoint system—bypass the need for high-order differentiation of noisy displacement data, using only first derivatives:

$$\frac{\partial J}{\partial\mu}(x) = \Re[2\,\nabla^s u_0(x) : \nabla^s \overline{v}(x)], \qquad \frac{\partial J}{\partial\eta_\mu}(x) = \Re[2\,i\omega\,\nabla^s u_0(x) : \nabla^s\overline{v}(x)]$$

A hybrid one-step Helmholtz-based initialization is crucial due to severe non-convexity. The ViscoReg reconstruction scheme demonstrates substantial improvements in modulus recovery, edge sharpness, and artifact suppression, particularly for localized reconstructions in regions of high signal-to-noise (Ammari et al., 2014).

3. Viscous Corrections in Rate-Independent Evolution Problems

ViscoReg designates the introduction of viscous penalties into time-incremental minimization schemes for abstract metric-topological systems $(X, \mathcal E, \mathsf d)$ driven by energy functionals and dissipation quasi-distances (Minotti et al., 2016). Standard energetic solutions correspond to global minimizers with respect to $d$, but the ViscoReg augmentation:

$$D(u,v) = d(u,v) + \delta(u,v), \quad \delta(u,v) = p\,d(u,v)^2$$

penalizes large jumps, localizing stability and refining the energy balance via extra jump cost terms.

The limiting curves (Visco-Energetic solutions) satisfy

• Localized stability: $$\mathcal{E}(t,u(t)) \leq \mathcal{E}(t,v) + d(u(t),v) + \delta(u(t),v)$$ outside jump times
• Refined energy balance: incorporating augmented total variation and jump penalties

This formulation interpolates between pure energetic and balanced-viscosity (BV) regimes, tightening jump transitions while maintaining broad applicability to convex, double-well, Allen–Cahn, and delamination problems (Minotti et al., 2016).

4. Viscosity-Augmented Control in 3D Viscoelastic Manipulation

ViscoReg further appears as a physics-based PDE model and control framework for compliant manipulation of viscoelastic materials in robotics (Ma et al., 11 Apr 2025). The central dynamic is the unified 3D viscoelastic PDE:

$$\frac{\partial\phi}{\partial t} = \epsilon\,\Delta\phi + a_1\,f + a_2\,\frac{\partial f}{\partial t} + \lambda\phi$$

where $\Delta\phi$ represents stress diffusion (Maxwell effect), $a_1 f$ (instant stiffness), $a_2 \partial_t f$ (damping), and $\lambda\phi$ (restoring force).

Material parameters are estimated in real time by a PDE observer driven by filtered regressor signals. Control is implemented via a dual-layer architecture:

• Outer loop: PD/admittance law updating reference deformation in response to force errors
• Inner loop: Reaction-diffusion PDE for deformation error with boundary Dirichlet constraints using backstepping kernels

Experimental results confirm sub-millimeter deformation accuracy and stable force tracking for a diverse set of soft and rigid objects, facilitating precision manipulation in soft robotics and surgical applications (Ma et al., 11 Apr 2025).

5. Stabilization of High-Weissenberg Number Viscoelastic Fluid Simulation

ViscoReg extends to high-Weissenberg number numerical stabilization for viscoelastic fluid simulations on unstructured meshes via the finite-volume method (Niethammer et al., 2017). The approach modularly combines rheological models (Oldroyd-B, PTT, Giesekus) with advanced stabilization methods:

• Change-of-variable representations: k-th root (RCR) and logarithm (LCR) of conformation tensors, ensuring SPD preservation and mitigation of High-Weissenberg Number Problem (HWNP)
• Semi-implicit stress interpolation correction (SISIC): Prevents checkerboard artifacts on co-located grids by lumping face-interpolated stress terms into the momentum matrix

The stabLib library allows runtime selection of transformation kernels and rheological models. Benchmarks on the planar $4{:}1$ contraction problem show robust performance and mesh-convergence far beyond classical HWNP limits (Wi up to $10^4$ for PTT) (Niethammer et al., 2017).

6. Comparative Features of ViscoReg Methodologies

Domain	Core ViscoReg Mechanism	Principal Benefit
Neural SDFs (Krishnan et al., 1 Jul 2025)	Decaying Laplacian penalization	Unique, stable geometric reconstructions
Viscoelastic imaging (Ammari et al., 2014)	Adjoint-based viscosity gradients	High-resolution modulus/viscosity maps
Rate-indep. systems (Minotti et al., 2016)	Viscous penalized minimization	Localized jumps, refined energy balances
Robotic manipulation (Ma et al., 11 Apr 2025)	3D viscoelastic PDE/observer/control	Precision compliant multi-material control
Fluid simulation (Niethammer et al., 2017)	CVR stabilizations, SISIC	Stable simulation at high Wi, modular design

The commonality is introducing viscous terms—whether Laplacian regularization, viscous jump penalties, or transformation kernels—to ensure uniqueness, stability, and physical realism in complex, high-dimensional inverse or evolution problems.

7. Theoretical and Empirical Impact

ViscoReg strategies are grounded in rigorous mathematical analyses of stability, uniqueness, and error bounds (e.g., $L^\infty$ generalization error for neural SDFs, Lyapunov results for observer-based control), and validated by numerical and experimental benchmarks. The approach resolves critical issues of ill-posedness and instability pervasive in both forward and inverse PDE problems, offering scalable, unifying methodologies that extend across computational imaging, mechanics, robotics, and numerical simulation. Future directions include further generalization to multi-physics inverse problems, integration with adaptive sampling, and real-time feedback control in complex, uncertain environments.