Jin and Liu state the exact mapping for nonlinear Hamilton-Jacobi equations and scalar nonlinear hyperbolic PDEs, for arbitrary nonlinearity. Systems of conservation laws lie outside that stated scope. Because the map is exact there is no convergence condition analogous to Carleman's $R < 1$; what the mapping preserves is physical observables rather than a directly readable solution vector.

$F$, $y_{\mathrm{in}}$, $T$, $\varepsilon$, and a truncation or lift parameter (Carleman truncation level $N$, a phase-space grid, the level-set dimension, the homotopy order).

1. Nonlinear initial-value problem → Linear ODE systemtrack the solution as a zero level set

   The lift is stated twice, once for each class the paper claims it for. For the $(d+1)$-dimensional Hamilton-Jacobi equation in gradient form, $\partial_t u^{[k]} + \nabla H(u^{[k]}, x) = 0$ with $u^{[k]} = \nabla S^{[k]} \in \mathbb{R}^d$, the level-set function $\phi_i^{[k]}(t,x,p)$ is defined by $\phi_i^{[k]}(t, x, p = u^{[k]}(t,x)) = 0$ for $i = 1, \ldots, d$ and satisfies the linear Liouville equation $\partial_t \phi^{[k]} + \nabla_p H \cdot \nabla_x \phi^{[k]} - \nabla_x H \cdot \nabla_p \phi^{[k]} = 0$ from $\phi_i^{[k]}(0,x,p) = p_i - u_i^{[k]}(0,x)$, in $2d+1$ dimensions. For the $(d+1)$-dimensional scalar hyperbolic equation $\partial_t u^{[k]} + F(u^{[k]}) \cdot \nabla_x u^{[k]} + Q(x, u^{[k]}) = 0$ the same device gives the linear transport equation $\partial_t \phi^{[k]} + F(p) \cdot \nabla_x \phi^{[k]} - Q(x,p)\,\partial_p \phi^{[k]} = 0$ from $\phi^{[k]}(0,x,p) = p - u_0^{[k]}(x)$, in $d+2$ dimensions, "the only difference" being that "now $p \in \mathbb{R}^1$ instead of being a $d$-dimensional vector". Neither lift truncates anything, so neither has a convergence parameter to mark: "no linear approximation is made. The mapping is exact", and the whole of the price is "at the expense of doubling the dimension". assumption: These two classes are the stated scope, and the paper draws the boundary itself: "this same result, however, cannot be done analytically for general nonlinear PDEs". A general nonlinear PDE is instead discretised in space first, into $D$ nonlinear ODEs with $D = d^2/\varepsilon$ under a Lagrangian discretisation or $D = (d/\varepsilon)^d$ under an Eulerian one, and only that system is given a linear representation. The solution comes back as a set rather than as a vector — $u^{[k]}(t,x) = \{p(t,x) \mid \phi_i^{[k]}(t,x,p) = 0,\ i = 1, \ldots, d\}$, the intersection of the $d$ zero level sets — and since $\phi_i^{[k]}(t,x,p) = 0$ may have $J_k$ roots $p_\gamma$, every branch $u_\gamma^{[k]}$ is kept. What the lift preserves is therefore an observable: for $G : \mathbb{R}^d \to \mathbb{R}$, Definition 11 sets $\langle G(t,x)\rangle = \int_{\mathbb{R}^d} G(p)\psi(t,x,p)\,dp = \frac{1}{M}\sum_{k=1}^{M}\sum_{\gamma=1}^{J_k} G(u_\gamma^{[k]}(t,x))/J_\gamma^{[k]}$, each branch weighted by the Jacobian $J_\gamma^{[k]} = |\det(\partial\phi^{[k]}/\partial p)|$ evaluated at $p = u_\gamma^{[k]}(t,x)$; taking $G(p) = 1, p, |p|^2$ gives the zeroth, first and second moments — density, momentum and kinetic energy in the semiclassical setting the paper works through — and the moments of $\psi$ give those same physical observables in the special case $A_0(x) = 1$. Those weights are the argument for encoding the lifted variable rather than $u$: recovering observables from amplitudes proportional to the original solutions "requires one to inject the Jacobian factors in Definition 11 explicitly", and those factors "are not generally a priori known since they depend on the solutions of the PDE and also cannot be ignored", whereas with a level-set encoded state "these Jacobian factors are automatically taken care of".

   assumption

A linear generator with any inhomogeneity, a lift map, a readout map, and an error bound as a function of the truncation parameter.

given  a (d+1)-dimensional nonlinear PDE with M initial data, k = 1..M:
           Hamilton-Jacobi in gradient form,
               d_t u^[k] + grad H(u^[k], x) = 0,        u^[k] = grad S^[k] in R^d
           or scalar hyperbolic,
               d_t u^[k] + F(u^[k]).grad_x u^[k] + Q(x, u^[k]) = 0,   u^[k] in R

# 1. lift: put the solution inside the zero level set of a new function

define  phi^[k](t, x, p)  by   phi_i^[k](t, x, p = u^[k](t,x)) = 0,   i = 1..d
    initial data:  phi_i^[k](0, x, p) = p_i - u_i^[k](0, x)          # Hamilton-Jacobi
                   phi^[k](0, x, p)   = p   - u_0^[k](x)             # scalar hyperbolic

# 2. the lifted equation is linear -- exactly, for arbitrary nonlinearity

Hamilton-Jacobi:    d_t phi + grad_p H . grad_x phi - grad_x H . grad_p phi = 0
                    # (2d+1)-dimensional
scalar hyperbolic:  d_t phi + F(p) . grad_x phi - Q(x,p) d_p phi = 0
                    # (d+2)-dimensional, because p is a scalar here
    # no truncation, and therefore no convergence parameter:
    # "the price is a higher-dimensional linear problem", not an error term
    # there is no analogue of Carleman's R < 1

# 3. read back

u^[k](t, x) = { p(t,x) | phi_i^[k](t,x,p) = 0, i = 1..d }
    # the intersection of the d zero level sets
    # phi_i^[k] = 0 may have J_k roots p_gamma: all multi-valued branches are kept

# 4. what the mapping preserves is physical observables, not a directly
#    readable solution vector

<G(t,x)> = integral_{R^d} G(p) psi(t,x,p) dp
         = (1/M) sum_{k=1..M} sum_{gamma=1..J_k} G(u_gamma^[k](t,x)) / J_gamma^[k]
    # J_gamma^[k] = |det(d phi^[k] / d p)| at p = u_gamma^[k](t,x)
    # the level-set encoding carries these Jacobian weights automatically;
    # an amplitude encoding of u itself would have to inject them, and they are
    # not known a priori

return  the linear PDE, its initial datum, and the observable rule

Stated in the abstract as complete-algorithm claims, not a standalone cost for the mapping: computing physical observables at a cost independent of $M$, the number of initial data, for arbitrary nonlinearity; up to exponential advantage in both the dimension of the PDE and the error in computing its observables, depending on the details of the initial data; and for general nonlinear PDEs, quantum advantage with respect to $M$ in the large-$M$ limit. No closed form appears in the abstract.