DInf-Grid: Neural, Optimization & Digital Grid

• DInf-Grid is a multifaceted framework integrating a neural feature grid for smooth PDE solving, distributed optimization for power network infeasibility, and a digital grid vision for market-integrated energy systems.
• The neural PDE paradigm uses multi-scale grids with Gaussian RBF interpolation to compute closed-form high-order derivatives and accelerate convergence over traditional MLP-based solvers.
• The power network and digital grid approaches employ distributed interior-point methods and cyber-physical integration to enhance grid resilience, efficiency, and real-time market operations.

DInf-Grid formally refers to three distinct but technologically significant paradigms, each anchored in recent scholarly contributions: (1) a neural feature grid for differential equation solvers (Kairanda et al., 15 Jan 2026), (2) a distributed convex-optimization framework for identifying infeasibility in combined transmission and distribution (T&D) power networks (Ali et al., 2024), and (3) a "Digital Grid" vision for a cyber-physical, market-integrated power system (Chakrabortty et al., 2017). Each usage shares a foundational emphasis on combining grid-centric data structures with advanced computational or optimization methodologies, but operates in distinct domains—machine learning for PDEs, power systems optimization, and digital infrastructure for energy markets. This article presents an integrated overview of all three DInf-Grid paradigms.

1. Infinitely Differentiable Neural Feature Grids

DInf-Grid, as introduced in (Kairanda et al., 15 Jan 2026), establishes a neural field representation for PDEs that combines the locality and speed of grid-based methods with smooth, closed-form differentiability. Unlike coordinate-based MLP solvers (e.g., SIREN), which are globally parameterized and computationally intensive, DInf-Grid employs multi-scale, co-located grids of feature vectors interpolated with Gaussian RBF kernels. This interpolation achieves $C^\infty$ spatial smoothness, enabling the computation of all high-order derivatives required by strong-form PDE residuals. The method is particularly notable for its use of stacked grids at dyadically decreasing resolutions, concatenating multi-scale features for each query.

The representation thus addresses two limitations of prior approaches: (i) it supports efficient, scalable training by restricting updates to local neighborhoods on the grid (as in Instant-NGP or K-Planes), and (ii) it maintains closed-form, stable derivatives via RBFs, which d-linear (trilinear, bilinear) interpolation methods cannot provide.

2. Mathematical Formulation and Differentiability

The core operation of the DInf-Grid representation at each scale $s$ is:

$f_s(x) = \sum_{x_j\in N_p(x)} w(x, x_j) F_s(x_j)$

with RBF weights

$$w(x, x_j) = \frac{\phi(\|x - x_j\|)}{\sum_{x_k\in N_p(x)} \phi(\|x - x_k\|)}$$

for $\phi(r) = \exp\left[-(\varepsilon r)^2\right]$. The complete feature vector $f(x) = (f_0(x), ..., f_{S-1}(x))$ is decoded via a shallow neural network $u(x) = d(f(x); \theta)$. Derivatives are analytically computable as $w$ admits closed-form gradients and higher derivatives due to the Gaussian kernel, so automatic differentiation yields $\nabla^k u(x)$ for any order $k$.

In a direct comparison, grid-based methods using d-linear interpolation are limited to $C^0$ continuity at cell boundaries and $C^1$ within interiors, causing $\nabla^2 u(x)$ and higher derivatives to be either identically zero or discontinuous, which renders such methods unsuitable for strong-form PDE residual losses.

3. Training Methodology and PDE Losses

Implicit training is performed by minimizing the residuals of the governing strong-form PDE via uniform (typically stratified) sampling:

$$L(F, \theta) = \int_{\Omega} \| \mathcal{F}(x, u, \nabla u, \nabla^2 u, ..., g(x)) \|^2 dx + \text{boundary/data terms}$$

with task-specific forms:

• Poisson image reconstruction: $$L_{\text{grad}} = \int_\Omega \|\nabla u(x) - \nabla g(x)\|^2 dx$$, $$L_{\text{lapl}} = \int_\Omega \|\Delta u(x) - \Delta g(x)\|^2 dx$$
• Helmholtz wavefields: $$L_{\text{helm}} = \int_\Omega |(\Delta + m\omega^2)u(x) + g(x)|^2 dx$$
• Kirchhoff-Love shell/cloth: $$L_{\text{cloth}} = \int_\Omega [W(\epsilon(u), \kappa(u); \lambda) - g(x) \cdot u(x)] dx$$

Dirichlet boundaries are enforced using a smooth blending: $u(x) \gets u(x)B(x) + h(x)(1 - B(x))$ with $$B(x) = 1 - \exp(-\text{dist}(x, \partial \Omega)/\delta)$$. Optimization is performed via Adam with scale- and task-adapted learning rates.

4. Performance, Benchmarks, and Ablation Analysis

DInf-Grid attains significant computational efficiency and accuracy relative to coordinate-based MLP and hybrid grid methods. In Poisson image reconstruction (512x512 RGB), DInf-Grid achieves $\sim$20$\times$ speedup over SIREN and +20 dB higher PSNR with only seconds to minutes of training. Unlike K-Planes, DInf-Grid supports Laplacian supervision because its second derivatives are nontrivial. For Helmholtz and Eikonal PDEs, DInf-Grid delivers stable, fast convergence where Instant-NGP and K-Planes fail due to their vanishing or discontinuous higher-order derivatives.

Ablation studies demonstrate that optimal RBF shape parameters ($\varepsilon \in [0.6, 1.0]$, $p=3$ neighbor rings) balance locality and smoothness. Multi-resolution grids ($S=4$) double or triple convergence rates of PDE residuals and PSNR against single-scale ($S=1$).

5. Distributed Infeasibility Analysis in Power Networks

In power systems engineering, DInf-Grid also refers to a distributed infeasibility analysis methodology for combined transmission (positive-sequence, AC) and distribution (three-phase) grids (Ali et al., 2024). This methodology couples a unified $I$-$V$ circuit-theoretic model with a distributed primal-dual interior point (PDIP) optimization to quantify and localize infeasibility (i.e., violations of Kirchhoff's current law and operational limits given unsolvable configurations typically caused by DER backfeed, high load, or grid faults).

Each component—transmission, distribution, and T&D interface—admits a precise algebraic model. Infeasibility sources (artificial current/voltage injections) are introduced at all buses, and an optimization minimizes the $\ell_p$-norm of these sources subject to power flow and operation constraints:

$$\min_{X, \mathcal{I}} \sum_{s_T \in S^T} \|\mathcal{I}^{T}_{s_T}\|_p^p + \sum_{s_D \in S^D} \sum_{\Omega=a,b,c} \|\mathcal{I}^{D}_{s_D, \Omega}\|_p^p$$

A Gauss–Jacobi–Newton (GJN) decomposition is employed for scalability, allowing privacy-preserving, distributed solution across subdomains, with KKT systems solved iteratively.

6. Digital Grid: Market-Integrated Cyber-Physical Power Systems

DInf-Grid is further used synonymously with the "Digital Grid" concept (Chakrabortty et al., 2017), envisioning a cyber-physical platform for open, high-frequency energy markets and control. This architecture integrates layered physical (energy) and cyber (information) infrastructures, including:

• Physical energy layer (transmission, distribution, DERs, smart transformers)
• Digital/cyber-information layer (PMUs, edge/cloud analytics, SDN/NFV-enabled communications)
• Real-time market design (wholesale, real-time, and transactive retail layers), employing coordinated market clearing, locational pricing, and autonomous DER dispatch.

Core mathematical models include multi-objective real-time market optimization, dynamic state-space grid control, and distributed ADMM for DER coordination. Case studies show that DInf-Grid operations stabilize voltage and frequency, reduce unserved energy, and improve DER utilization dramatically on modern microgrids and campus-scale VPPs.

7. Limitations and Future Directions

Across all paradigms, DInf-Grid inherits specific technical and operational challenges:

• Neural feature grid (PDEs): Curse of dimensionality in $d\geq4$, boundary artifacts (Runge's phenomenon), sensitivity to RBF shape parameter $\varepsilon$, and computational overhead from evaluating RBF interpolants.
• Power network infeasibility analysis: Distributed methods require careful tuning and synchronization, and $\ell_2$ objectives can obscure localization, with $\ell_1$ needed for sparse correction.
• Digital grid cyber-physical systems: High integration between market, control, and cyber-security layers increases system complexity and potential interoperability risks.

Proposed future work includes adaptive, learned RBF kernels, manifold projection for high-dimensional PDEs, customized parallel kernels for RBF lookup, data-driven cyber-defense, and fault-resilient optimization for distributed power networks.

• DInf-Grid: A Neural Differential Equation Solver with Differentiable Feature Grids, (Kairanda et al., 15 Jan 2026)
• Distributed Primal-Dual Interior Point Framework for Analyzing Infeasible Combined Transmission and Distribution Grid Networks, (Ali et al., 2024)
• Digital Grid: Transforming the Electric Power Grid into an Innovation Engine for the United States, (Chakrabortty et al., 2017)