---
title: 'Zubov-Net: Neural Methods for ROA Estimation'
url: https://www.emergentmind.com/topics/zubov-net
type: topic
---

# Zubov-Net: Neural Methods for ROA Estimation

Searching arXiv for the specified papers and closely related work on Zubov-Net.
Zubov-Net is a label used in recent literature for a family of neural-network methods derived from Zubov theory for nonlinear dynamical systems. Across these works, the central object is a learned scalar function—variously denoted \(W\), \(\tilde W_\theta\), \(V_\phi\), or \(W_\theta\)—whose sublevel geometry encodes a region of attraction (ROA), a safe domain of attraction, or a safe-stabilizable domain. The common technical motif is to approximate either a Zubov partial differential equation, a Zubov–Hamilton–Jacobi–Bellman equation, or a Zubov–Koopman fixed-point relation, and then use the learned function for controller synthesis, safety certification, robustness regularization, or ROA estimation [2506.01356][2511.09523][2604.00941][2509.21879][2311.15119].

## 1. Terminological scope and problem setting

In the cited literature, “Zubov-Net” does not denote a single canonical architecture. Instead, it names several related constructions that all use Zubov-style characterizations of attraction domains. One line of work presents a two-stage framework that jointly synthesizes a neural state-feedback controller \(u_\theta(x)\) and a Lyapunov-like function \(V_\phi(x)\) for continuous-time systems, with direct ROA estimation and formal verification via an extended \(\alpha,\!\beta\)-CROWN verifier [2506.01356]. A second line treats “Zubov-Net” as a physics-informed neural network (PINN) for solving a Dirichlet-form Zubov PDE on a safe set boundary, so that the resulting function acts as a Lyapunov-barrier certificate [2511.09523]. A third line extends the underlying PDE viewpoint to control-affine safe stabilization through a Zubov–HJB formulation whose viscosity solution yields a maximal control Lyapunov-barrier function (CLBF) [2604.00941]. A fourth line uses the name for an adaptive stable learning framework for Neural ODE classifiers, where Zubov’s equation is reformulated as a consistency condition between prescribed regions of attraction (PRoAs) and true ROAs [2509.21879]. A fifth, earlier line uses the term for a neural approximation of a Zubov–Koopman operator, learned from data and iterated to recover a Zubov solution for an unknown system [2311.15119].

The unifying task is the constructive approximation of a function whose level sets separate asymptotically stable behavior from instability or unsafety. In the classical autonomous setting, the exact ROA is identified with \(\{x:W(x)<1\}\); in the safety-constrained setting, the same structure is applied on a safe set \(D\) with Dirichlet data \(W=1\) on \(\partial D\); in the control-affine setting, a maximization over admissible controls enters the PDE; and in the Neural ODE classification setting, each class equilibrium \(c_i\) is equipped with its own \(W_i\), whose sublevel set \(D_{W_i}\) is interpreted as a PRoA [2506.01356][2511.09523][2604.00941][2509.21879].

## 2. Zubov-theoretic foundations

For continuous-time closed-loop dynamics
\[
\dot x = f(x), \qquad f(0)=0,
\]
Zubov’s theorem characterizes the true ROA \(\mathcal A\subset\mathbb R^n\) by a continuously differentiable function \(W:\mathcal A\to[0,1)\) such that \(W(0)=0\), \(0<W(x)<1\) for \(x\neq 0\), \(W(x)\to 1\) as \(x\to\partial\mathcal A\) or \(\|x\|\to\infty\), and
\[
\nabla W(x)^\top f(x)=-\Psi(x)(1-W(x)).
\]
A practical choice reported from Liu et al. is
\[
\Psi(x)=a(1+W(x))\|x\|^p,
\]
which yields
\[
\nabla W(x)^\top f(x)=-a(1+W(x))(1-W(x))\|x\|^p.
\]
Imposing \(W(x)=1\) on \(\partial\mathcal A\) pins down the maximal ROA exactly as \(\{x:W(x)<1\}\) [2506.01356].

An equivalent autonomous formulation writes
\[
\nabla W(x)\cdot f(x)=-\eta(x)(1-W(x)),
\]
with \(W(0)=0\) and \(W(x)\to 1\) on \(\partial D\), where \(D\) is the domain of attraction. In that representation, \(V=1-W\) satisfies
\[
\nabla V(x)\cdot f(x)=-\eta(x)V(x),\qquad V(0)=1,\quad V\to 0 \text{ on }\partial D,
\]
and the ROA is exactly
\[
D=\{x:W(x)<1\}=\{x:V(x)>0\}.
\]
This is the form used in the Zubov–Koopman lifting approach [2311.15119].

For safety-constrained autonomous systems, the literature introduces a Dirichlet-form Zubov PDE on the safe domain \(D=\{x:h(x)<1\}\):
\[
-\,\nabla W(x)\cdot f(x)+(1-W(x))\,\varphi\!\bigl(\beta(V(x))\bigr)\,\omega(x)=0,
\]
with boundary conditions
\[
W(0)=0,\qquad W(x)=1\quad \forall x\in\partial D.
\]
On a larger compact ROI \(\Omega\supset D\), a positively rescaled modified problem is written as
\[
\begin{cases}
-\,\nabla \tilde W(x)\cdot\tilde f(x)
+\bigl(1-\tilde W(x)\bigr)\,\varphi\bigl(\beta(\tilde V(x))\bigr)\,\|x\|^2=0,\\[0.5ex]
\tilde W(0)=0,\qquad \tilde W(x)=1\quad \forall x\in\partial\Omega,
\end{cases}
\]
where \(\tilde f(x)=f(x)(1-h(x))^k/\theta(x)\) [2511.09523].

For control-affine systems
\[
\dot x=f(x)+g(x)u,
\]
the Zubov construction becomes an HJB-type PDE. If \(V(x)\) is the value function defined by an infinite-horizon running cost \(\eta(x,u)\), then \(V\) is the unique viscosity solution of
\[
-\,\max_{u\in U}\{\nabla V(x)\cdot[f(x)+g(x)u]+\eta(x,u)\}=0
\]
on the maximal safe-stabilizable domain \(D\), with \(V(0)=0\) and \(V(x)\to\infty\) as \(x\to\partial D\) or \(\|x\|\to\infty\). Under the Zubov transformation \(W=\beta(V)\), one obtains
\[
-\,\max_{u\in U}\{\nabla W(x)\cdot[f(x)+g(x)u]+(1-W(x))\phi(W(x))\eta(x,u)\}=0,
\]
with \(W(0)=0\) and \(W(x)=1\) on \(\partial D\) [2604.00941].

These formulations all preserve the same geometric principle: if the exact solution is obtained, then \(\{W<1\}\) is the maximal attractor-compatible domain. This suggests that the principal source of variation among Zubov-Net methods lies not in the underlying theorem, but in how the equation is approximated, sampled, regularized, and verified.

## 3. Two-stage neural controller synthesis and formal certification

The controller-synthesis variant parameterizes both a controller and a Zubov function by multilayer perceptrons. The controller is
\[
u_\theta(x)=
c\;\tanh\!\Bigl(\mathrm{NN}_c(x)-\mathrm{NN}_c(0)+\tanh^{-1}\!\bigl(\tfrac{u^*}{c}\bigr)\Bigr),
\]
where \(c\) is the known symmetric input limit and \(u^*\) is the equilibrium input. Each output coordinate is individually clamped to \([-c,c]\). The Zubov network is
\[
V_\phi(x)=\sigma(\mathrm{NN}_v(x)),
\]
with \(\sigma(z)=\tfrac{1}{1+e^{-z}}\), ensuring \(V_\phi(x)\in(0,1)\). In practice, both \(\mathrm{NN}_c\) and \(\mathrm{NN}_v\) are standard fully connected layers with \(3\)–\(5\) hidden layers, width \(64\)–\(256\), tanh or ReLU hidden activations, and no batch normalization [2506.01356].

Training proceeds in two stages. Stage 1 performs ROA estimation through a Zubov-guided curriculum. Every \(\gamma\) steps, an UpdateDomain procedure expands the current domain \(\Omega\). Two batches of \(N\) points are sampled by “Zubov-guided PGD”: interior points minimize \(\mathrm{ReLU}(V_\phi(x)-c)\), and exterior points minimize \(|V_\phi(x)-1|\). Boundary points are sampled on \(\partial(2\Omega)\). The composite loss includes \(L_{\text{zero}}\), a Zubov PDE residual \(L_{\text{pde}}\), a trajectory-consistency term \(L_{\text{data}}\), a controller term \(L_{\text{ctrl}}\), and a boundary term \(L_{\text{bnd}}\), with weights learned via uncertainty weighting. The UpdateDomain procedure samples \(N_T\) points in \(\{x:0<V_\phi(x)<c\}\) by PGD, simulates them for time \(T\) at step \(\Delta t\), records dimensionwise minima and maxima of convergent trajectories, forms a new box \(\Omega_d=[d_{\min},d_{\max}]\), and expands it by factor \(\beta>1\) [2506.01356].

Stage 2 is a CEGIS refinement stage. It maintains a small buffer of counterexamples, attacks points in \(\Omega\) via PGD to maximize
\[
L_{\mathrm{cex}}(x)=\min\bigl(\nabla V_\phi(x)^\top f(x),\,V_\phi(x)-c',\,c-V_\phi(x)\bigr),
\]
and then descends
\[
L_{\mathrm{cegis}}(\theta)=\mathrm{ReLU}\bigl(\nabla V_\phi(x_{\mathrm{cex}})^\top f(x_{\mathrm{cex}})\bigr)
\]
plus a regularizer
\[
L_{\mathrm{reg}}=\bigl|V_\phi(0)-v_0\bigr|+\sum_{y\in\partial\Omega}|V_\phi(y)-1|.
\]
Training stops when no new counterexamples appear. The stated intuition is that Stage 1 rapidly learns a roughly correct ROA by combining a physics-informed Zubov PDE loss with interior/exterior sampling and domain expansion, and Stage 2 removes local Lyapunov violations inside the estimated level set [2506.01356].

Formal certification is based on an extension of \(\alpha,\!\beta\)-CROWN to continuous-time Jacobian products. To certify forward invariance of \(V_\phi^{\le c_1}\), the method uses a theorem requiring, for some \(0<c_1<c_2\), negativity of \(\nabla V_\phi(x)^\top f(x,u_\theta(x))\) on \(\{c_1\le V_\phi(x)\le c_2\}\cap\Omega\) and inward-pointing vector field behavior on \(\partial\Omega\cap\{V_\phi\le c_2\}\). The verifier adds tight linear relaxations for \(\tanh'(z)=1-\tanh^2(z)\) and \(\sigma'(z)=\sigma(z)(1-\sigma(z))\) over input ranges \([l,u]\), supports these in the bound-propagation engine, and replaces repeated bisection on \(c_1,c_2\) with an adaptive branch-and-bound scheme that updates \(c_1,c_2\) when a small counterexample is found, without re-verifying already certified subdomains [2506.01356].

Reported quantitative results are explicit. ROA volume improvements over baselines are up to \(5\times\) on \(2\)D systems, \(330\times\) on Cartpole (\(4\)D), \(380\times\) on PVTOL (\(6\)D), and \(1.5\times 10^5\times\) on a \(2\)D quadrotor, with an overall estimate of up to \(1.17\times 10^9\) via Monte Carlo in \(12\)D. On \(2\)D benchmarks, dReal takes \(113\)–\(39\,265\) s, whereas the extended CROWN verifier finishes each in \(3\)–\(4\) s, corresponding to a \(40\times\)–\(10\,000\times\) speedup. The same source states that dReal often fails on Cartpole, while CROWN verifies it in approximately \(76\,000\) s. It also notes that beyond \(6\)D, Jacobian bound tightness remains a challenge; in \(6\)D, forward invariance is verified only for a thin band, and empirical PGD or Monte Carlo is used for higher dimensions [2506.01356].

## 4. Dirichlet Zubov-Net for safety and maximal CLBFs

The safety-oriented PINN formulation learns a function \(\tilde W_\theta\) on a compact ROI \(\Omega\) using a fully connected feed-forward network with two hidden layers of width \(30\) and smooth activations such as tanh. Training uses \(N\) collocation points in \(\Omega\setminus\partial\Omega\), \(M\) boundary points on \(\partial\Omega\), and the origin. The losses are
\[
\ell_{\rm PDE}(\theta)
= \frac1N\sum_{i=1}^N
\Bigl|
-\,\nabla \tilde W_\theta(x_i)\cdot\tilde f(x_i)
+(1-\tilde W_\theta(x_i))\,\varphi\bigl(\beta(\tilde V(x_i))\bigr)\,\|x_i\|^2
\Bigr|^2,
\]
\[
\ell_{\rm bdy}(\theta)=\frac1M\sum_{j=1}^M|\tilde W_\theta(x_j^{\rm bdy})-1|^2,
\qquad
\ell_{\rm origin}(\theta)=|\tilde W_\theta(0)|^2,
\]
and the total loss is
\[
\mathcal L(\theta)=\ell_{\rm PDE}(\theta)+\lambda_{\rm bdy}\ell_{\rm bdy}(\theta)+\lambda_0\ell_{\rm origin}(\theta)+\lambda_{\rm reg}\|\theta\|^2.
\]
The reported hyperparameters are \(\lambda_{\rm bdy}=1\), \(\lambda_0=100\), \(\lambda_{\rm reg}=10^{-6}\), with Latin-Hypercube sampling for collocation points, uniform sampling on \(\partial\Omega\), \(N=3{,}000\), \(M=3{,}000\), and Adam for \(20\) epochs (\(300\)k gradient steps) at learning rate \(10^{-3}\) [2511.09523].

Verification in this framework targets both forward invariance and obstacle avoidance. For a sublevel set
\[
\mathcal S_c=\{x\in\Omega:\tilde W_\theta(x)\le c\},\qquad c<1,
\]
the goal is to prove
\[
\nabla \tilde W_\theta(x)\cdot f(x)<0 \quad \forall x\in\mathcal S_c,
\qquad
\mathcal S_c\cap U=\emptyset.
\]
LyZNet calls dReal or Z3 to prove the first-order formulas
\[
\forall x,\quad
[\tilde W_\theta(x)\le c\wedge h(x)<1]
\Longrightarrow
\nabla \tilde W_\theta(x)\cdot f(x)<-\varepsilon,
\]
and
\[
\forall x,\quad
[\tilde W_\theta(x)\le c]
\Longrightarrow
h(x)<1,
\]
for some small margin \(\varepsilon>0\). Under the corresponding lemma, if \(\nabla \tilde W_\theta(x)\cdot f(x)<0\) on \(\mathcal S_c\) and \(\mathcal S_c\cap U=\emptyset\), then \(\mathcal S_c\) is forward-invariant and contains no unsafe points [2511.09523].

This safe-domain formulation is closely related to the later control-affine Zubov–HJB theory. There, the maximal safe-stabilizable domain is
\[
D=\{x:\exists\,u(\cdot)\in L^\infty \text{ s.t. } \phi(t;x,u)\in S \ \forall t\ge 0 \text{ and } \phi(t;x,u)\to 0 \text{ as } t\to\infty\},
\]
under a compatibility assumption near the safe boundary. The transformed solution \(W=\beta\circ V\) is shown to be a continuous viscosity solution and a nonsmooth CLBF, with \(\{W<1\}=D\). Feedback is synthesized pointwise by
\[
u^*(x)\in \arg\max_{u\in U}\{-\nabla V(x)\cdot[f(x)+g(x)u]-\eta(x,u)\},
\]
or equivalently in Zubov form using \(\nabla W\). The associated “Zubov-Net” implementation recipe minimizes interior PDE residual, Dirichlet boundary losses, optional inequality penalties, and a gradient regularizer, using a fully connected MLP with tanh or softplus and either a sigmoid output or a clamped parameterization to keep \(W\in[0,1]\) [2604.00941].

The empirical benchmark reported for the Dirichlet safe-domain PINN is the reversed Van der Pol system with two circular obstacles, where the learned and verified level set recovered roughly \(98\%\) of the true area, while a quadratic CLF approach captured \(70\%\). Training times are reported as \(5\)–\(10\) minutes and formal verification as \(5\)–\(15\) minutes per example. The same source identifies two limitations: the need to choose a proper indicator \(\omega\) such that \(\omega\to\infty\) at \(\partial D\) and \(\int \omega\,dt\) diverges, and the fact that formal verification can be conservative and does not scale beyond \(n=4\)–\(5\) [2511.09523].

## 5. Zubov-Net for Neural ODE robustness and prescribed attraction geometry

A distinct use of the name appears in Neural ODE classification, where Zubov-Net is an “adaptive stable learning framework” that seeks to reconcile robustness and accuracy by actively controlling basin geometry. The system is
\[
\dot h(t)=f(h(t);\theta_f),\qquad h(0)=h_0\in H\subset\mathbb R^{d_h},
\]
with equilibrium set \(\mathcal A=\{c_1,\dots,c_L\}\), one equilibrium per class. The true ROA of equilibrium \(c_i\) is
\[
\mathcal D_f(c_i)=\{h\in H:\lim_{t\to\infty}\|h(t)-c_i\|=0\text{ when }h(0)=h\},
\]
and the PRoA is defined by a learnable Lyapunov function \(W_i:H\to[0,1]\) through
\[
D_{W_i}=\{h\in H:W_i(h)<1\}.
\]
The stated interpretation is that \(D_{W_i}\) is the current guess of \(\mathcal D_f(c_i)\) [2509.21879].

The central reformulation is a consistency loss derived from Zubov’s equation. If \(W_i(c_i)=0\), \(W_i(h)\to 1\) on \(\partial\mathcal D_f(c_i)\), and
\[
\dot W_i(h)=\frac{dW_i(h)}{dh}f(h)=-\Phi_i(h)(1-W_i(h))
\]
for all \(h\in D_{W_i}\), with \(\Phi_i(h)>0\) for \(h\neq c_i\), then \(D_{W_i}=\mathcal D_f(c_i)\). The paper turns this into
\[
l_{\rm con}^i(h)=\left\|\frac{dW_i(h)}{dh}f(h;\theta_f)+\Phi_i(h)(1-W_i(h))\right\|_2^2,
\]
and globally enforces
\[
\mathcal L_{\rm con}(\theta_f,\theta_W)=\sum_{i=1}^L \sup_{h\in D_{W_i}} l_{\rm con}^i(h).
\]
It states that \(\mathcal L_{\rm con}=0\) if and only if each PRoA exactly satisfies Zubov’s PDE and hence aligns with the true \(\mathcal D_f(c_i)\) [2509.21879].

Two additional losses control class assignment and basin separation. The classification term uses the Lyapunov-value vector
\[
\mathbf W(h)=\bigl[W_1(h),\dots,W_L(h)\bigr]^\top
\]
and the normalized inverse-value map
\[
\tilde\psi(h;C_\psi)=
\frac{\mathbf W(h)^{\circ-1}-\alpha\mathbf 1}
{\mathbf 1^\top(\mathbf W(h)^{\circ-1}-\alpha\mathbf 1)},
\qquad \alpha\in[0,1),
\]
with
\[
\mathcal L_{\rm cla}=-\log[\tilde\psi(h_x(T);C_\psi)]_y.
\]
The separation term samples boundary points \(\{h_j^{(i)}\}_{j=1}^m\subset\partial D_{W_i}\) and penalizes
\[
\mathcal L_{\rm sep}(\theta_W)=
\sum_{i=1}^L\sum_{j=1}^m\sum_{k\neq i}-W_k^\beta(h_j^{(i)}),
\qquad
W_k^\beta(h)=1-\exp(-\beta\,\mathcal V_k(h)),
\]
where \(W_k=1-e^{-\mathcal V_k}\) and \(\beta\in(0,1)\). The total objective combines classification, a term \(\lambda_1\mathcal L_{\rm FC}\), consistency, and separation [2509.21879].

A notable architectural contribution is the input-attention convex neural network for the pre-Lyapunov potential
\[
\mathcal V(x,c)=\sigma(g(x-c,c)-g(0,c))+\delta\|x-c\|_2^2.
\]
The \(k\)-layer IACNN uses softmax attention over \(x\) and previous features \(z_i\), with elementwise nonnegative \(A_i^{(z)}\) and convex non-decreasing \(\sigma_i\), so that \(g(\cdot,c)\) is convex in \(x\). The paper states that the softmax attention mechanism focuses on equilibrium-relevant features and also serves as weight normalization to maintain training stability in deep architectures [2509.21879].

The parallel boundary sampling algorithm updates radial lengths \(s_i\) along directions \(q_i\) by sign feedback until the target level \(\rho\approx 1\) is reached. All updates are performed in parallel, and the argument relies on radial monotonicity induced by strong convexity. The stated theoretical guarantees are: \(\mathcal L_{\rm con}=0\) if and only if \(\bigcup_i D_{W_i}=\mathcal D_f(\mathcal A)\); \(\mathcal L_{\rm con}=0\) implies \(D_{W_i}\cap D_{W_j}=\emptyset\) for \(i\neq j\); and if \(\mathcal L_{\rm con}=0\) and \(h_x(T)\in D_{W_y}\), then the entire trajectory \(h_x(t)\) for \(t\in[0,T]\) stays in \(D_{W_y}\) [2509.21879].

The empirical results are reported on SVHN, CIFAR-10, and CIFAR-100. For noise robustness, average over eight corruptions plus clean is \(87.48\%\) on SVHN versus \(87.24\%\) for the next best SODEF, \(80.00\%\) on CIFAR-10 versus \(78.89\%\), and \(37.86\%\) on CIFAR-100 versus \(36.33\%\). For white-box adversarial robustness, averaged over FGSM, PGD, BIM, APGD, and Jitter at \(\epsilon=8/255,16/255\), the results are \(74.30\%\) on SVHN versus \(71.01\%\) for FxTS-Net, \(57.12\%\) on CIFAR-10 versus \(50.95\%\), and \(22.80\%\) on CIFAR-100 versus \(22.40\%\). Clean accuracy remains on par or better than baselines, with CIFAR-10 reported as \(91.29\%\) versus \(91.45\%\) for a standard Neural ODE. The ablation claims are explicit: removing \(\mathcal L_{\rm sep}\) sharply degrades adversarial robustness, removing \(\mathcal L_{\rm con}\) degrades noise robustness, and the full tripartite loss gives the best overall trade-off [2509.21879].

## 6. Zubov–Koopman lifting and data-driven ROA learning

The Zubov–Koopman line addresses unknown autonomous systems from data. For
\[
\dot x=f(x),\qquad x\in\mathbb R^n,
\]
with asymptotically stable equilibrium at \(x=0\), the one-step Zubov–Koopman operator is defined for any bounded continuous \(h\) by
\[
(U_\tau h)(x)=\exp\!\Bigl(-\!\int_0^\tau \eta(\phi(s,x))\,ds\Bigr)\,h(\phi(\tau,x)),
\]
where \(\phi(t,x)\) is the flow and \(\eta\) is a positive-definite weight. The family \(\{U_t\}\) is a strongly continuous linear semigroup on \(C_b(\mathbb R^n)\), with infinitesimal generator
\[
Lh(x)=\nabla h(x)\cdot f(x)-\eta(x)h(x).
\]
The Zubov solution \(W\), extended by \(W\equiv 1\) outside the ROA, is the unique bounded fixed point of each one-step map:
\[
U_\tau W=W.
\]
This converts ROA estimation into operator learning [2311.15119].

The neural parameterization chooses feature functions
\[
\psi(x)=[\psi_1(x),\dots,\psi_N(x)]\in\mathbb R^N
\]
and approximates
\[
(U_\tau\psi)(x)\approx \psi(x)A,
\]
with \(A\in\mathbb R^{N\times N}\). A decoder row vector \(c\in\mathbb R^N\) then yields
\[
W_\theta(x)=c\,\psi(x).
\]
Under power iteration, the full ANN is
\[
W_\theta:x\mapsto c\bigl(\psi(x)A^k\bigr),
\]
which the source calls the “Zubov-Net.” Typical choices include sinusoidal, Fourier, monomial, or small-MLP features; \(A\) may be unconstrained or regularized; and \(c\) may use softplus to enforce \(W\in[0,1]\) [2311.15119].

Training combines an EDMD-style operator-fit term and a PDE residual. Given pairs \((x_i,\phi(\tau,x_i))\) and weights \(\exp(-\int_0^\tau \eta)\), the losses are
\[
\mathcal L_{\rm EDMD}=\frac1M\|\Psi A-\Pi\|_F^2+\alpha\|A\|_F^2,
\]
and
\[
\mathcal L_{\rm PDE}=
\frac1M\sum_i
\left\|
\nabla(c\,\psi(x_i))\cdot f(x_i)+\eta(x_i)(1-c\,\psi(x_i))
\right\|^2,
\]
with total loss
\[
\mathcal L=\mathcal L_{\rm EDMD}+\beta\,\mathcal L_{\rm PDE}.
\]
The recovery step uses
\[
W^{(k)}(x)=c\,\psi(x)\,A^k,
\]
and under a spectral gap assumption on \(A\), \(W^{(k)}\to W\) as \(k\to\infty\). The final Lyapunov candidate is \(V=1-W\) [2311.15119].

The paper states several convergence ingredients: the semigroup property \(U_{t+s}=U_t\circ U_s\); the improper-integral formula
\[
W(x)=1-\exp\!\left(-\int_0^\infty \eta(\phi(s,x))\,ds\right)
\]
yielding a bounded continuous function; uniqueness of the bounded viscosity solution of the Zubov PDE by a comparison principle; finite-rank approximation of \(U_\tau\) by projection onto leading eigenfunctions; EDMD strong convergence with sufficiently rich dictionary and enough data; and convergence of \(A^k\) to the fixed-point projector when the learned \(A\) has dominant eigenvalue \(1\) and spectral gap [2311.15119].

The numerical examples include reversed Van der Pol, a polynomial stiff system, a \(3\)D nonlinear system, and a two-machine power system with a sine nonlinearity. The common setup uses a region \(R\subset\mathbb R^n\) containing the ROA, \(\tau\approx 1\), \(\eta(x)=\|x\|\) or a minor variant, a small MLP dictionary with two hidden layers of \(32\) units and ReLU or smooth activations, batch size \(M=2000\)–\(10{,}000\), and \(500\)–\(1000\) epochs of joint EDMD plus PDE loss followed by \(10\)–\(20\) power iterations. In the reversed Van der Pol example, the Hausdorff error is approximately \(2\times 10^{-2}\). The report states that all examples converge in under \(20\) power iterations and are finally checked by verifying \(\nabla W\cdot f+\eta(1-W)\le 0\) on a dense grid in \(R\) [2311.15119].

## 7. Limitations, misconceptions, and research directions

A common misconception is that Zubov-Net denotes a single fixed neural architecture. The literature instead supports a broader interpretation: the term refers to several neural approximators of Zubov-type objects—PDE solutions, viscosity solutions, Lyapunov-barrier functions, or operator fixed points—adapted to different tasks such as controller synthesis, safety certification, Neural ODE robustness, or operator learning [2506.01356][2511.09523][2509.21879][2311.15119].

Another misconception is that all Zubov-Net methods are verification-complete. The sources distinguish sharply between approximate learning and formal certification. The controller-synthesis framework extends \(\alpha,\!\beta\)-CROWN and reports substantial speedups over dReal, but also states that Jacobian bound tightness remains a challenge beyond \(6\)D and that empirical PGD or Monte Carlo is used in higher dimensions [2506.01356]. The safe-domain PINN framework uses SMT solvers through LyZNet, yet explicitly notes that formal verification is conservative and does not scale beyond \(n=4\)–\(5\) [2511.09523]. The Neural ODE robustness framework proves consistency, non-overlap, and finite-time containment under the exact-loss condition \(\mathcal L_{\rm con}=0\), but its empirical claims concern robustness metrics rather than formal reachability certificates [2509.21879]. The Koopman-lifting approach provides strong convergence statements under regularity, richness, and spectral assumptions, but its final verification step is described as dense-grid checking rather than theorem-proving [2311.15119].

The main technical bottlenecks are also consistent across the literature. High-dimensional verification is limited by bound tightness for Jacobian terms in continuous-time systems [2506.01356]. Boundary-layer resolution may require many collocation points in very high dimensions [2511.09523]. Safe-domain formulations require a proper indicator \(\omega\) whose blow-up behavior is suitable near the boundary [2511.09523]. Control-affine formulations rely on compatibility assumptions near the safe boundary and on viscosity-solution machinery, which suggests that numerical realization is naturally more delicate than the autonomous case [2604.00941].

Taken together, these works position Zubov-Net not as a single algorithm, but as a research program centered on constructive approximations of maximal attraction-compatible functions. The consistent structural claim is that, when the relevant Zubov-type equation is solved accurately enough and the resulting certificate is formally or empirically validated, the sublevel set \(\{W<1\}\) provides a non-conservative under-approximation—or in the ideal exact case, the exact characterization—of the underlying ROA, safe domain of attraction, or safe-stabilizable region [2506.01356][2511.09523][2604.00941].

Source: https://www.emergentmind.com/topics/zubov-net