Low Residuals, Wrong Value: Learning the Viscosity Solution of Hamilton-Jacobi-Isaacs Equations with PINNs

Raul Jimenez; Pablo Tejerina-Pérez; Leonid Sarieddine; Pedro Tarancón-Álvarez; Marisol Traforetti; Pau Solé-Vilaró; Oleksii Ignatenko; Nazar Tkach

math.NAOct 19, 2026

Low Residuals, Wrong Value: Learning the Viscosity Solution of Hamilton-Jacobi-Isaacs Equations with PINNs

Authors: Raul Jimenez, Pablo Tejerina-Pérez, Leonid Sarieddine, Pedro Tarancón-Álvarez, Marisol Traforetti, Pau Solé-Vilaró, Oleksii Ignatenko, Nazar Tkach,

Organizations: Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona, Barcelona, Spain · Institució Catalana de Recerca i Estudis Avançats (ICREA), Barcelona, Spain · Departament de Física Quàntica i Astrofísica, Universitat de Barcelona, Barcelona, Spain · Faculty of Applied Sciences, Ukrainian Catholic University, Lviv, Ukraine ·

Abstract

Physics-informed neural networks (PINNs) and other residual-trained solvers judge a candidate solution by the size of its sampled PDE residual. For first-order Hamilton-Jacobi equations this is not sufficient: the same equation and boundary data admit Lipschitz almost-everywhere solutions that differ from the viscosity solution by $O(1)$ while violating the equation only on sets of measure zero. We study whether residual minimization selects such solutions in a fixed-horizon pursuit-evasion game whose Hamilton-Jacobi-Isaacs value function is known in closed form on the circle $S^1$ and on the flat tori $\mathbb{T}^{2}$ and $\mathbb{T}^{4}$ . We train four neural parameterizations with ten seeds on each geometry, 120 runs in total. A standard network with periodic Fourier features converges to folded almost-everywhere solutions in all 30 runs: small residuals and the expected capture-layer gradient, yet values that are wrong over most of the domain. A network with periodicity imposed through the loss recovers the viscosity solution in one and two dimensions but becomes unreliable in four. Two parameterizations that receive the geodesic distance do substantially better: a single network fed the wrapped displacement and its distance recovers the viscosity solution in all 30 runs, while a minimum over periodic image charts succeeds in 27 of 30 and introduces a wrong-image failure mode. Finally, for the convex Hamiltonian considered here, we use the semiconcavity of the viscosity solution to build a mesh-free finite-difference test that needs only forward network evaluations; it detects the folded solutions and the harmful min-of-charts failures without access to the exact value function. A small sampled residual thus does not by itself validate a learned Hamilton-Jacobi solution; the parameterization and independent structural diagnostics can be essential for selecting the viscosity solution.

Figures & tables

Figure 1: The value function $V^{*}$ of Eq. ( 7 ) for $\varepsilon=0.08$ , $\Delta v=0.2$ . Left: $V^{*}$ on $\mathbb{T}^{2}$ in the $(d,\tau)$ plane. We show the capture set (black band), the cone $d_{c}=\varepsilon+\Delta v\,\tau$ (green line), the capture crease $d=\varepsilon$ (a convex kink on the boundary), the horizon at which the cone reaches the point of $\mathbb{T}^{2}$ farthest from the pursuer (dotted line at $\tau=6.67$ ), the nearest points of the cut locus (dashed $d=1$ ) and the distances spanned by the cut locus (hatched band). Right: A time-slice $V^{*}(\bm{\Delta},\tau=4)$ on the fundamental domain of $\mathbb{T}^{2}$ . The dashed, red circle marks the cone $d_{c}=\varepsilon+\Delta v\,\tau$ ; inside it the value function grows linearly with $d$ , while outside it is $V^{*}=\tau$ .

Figure 2: The chart selector $m(\bm{\Delta},\tau)$ on $\mathbb{T}^{2}$ , drawn on $[0,2)^{2}$ with the pursuer at the origin $(0,0)$ and its periodic-images at the other corners. The faces defining the cut locus of the pursuer on the torus, which are $\tilde{\Delta}_{i}=\pm 1$ on the fundamental domain, appear as the midlines $\Delta_{i}=1$ . Left. Analytic nearest-image index $m^{*}(\bm{\Delta},\tau)$ . The four colors index each of the charts. Arrows show the optimal direction of motion of the pursuer, from the nearest periodic image, towards the evader. The charts meet on the ridge (cut locus). In the fundamental domain $[-1,1)^{2}$ (coordinates of the text), the nearest image is always in the chart with $\bm{b}_{m}=0$ . In the representative coordinates of the figure $\bm{\Delta}\in[0,2)^{2}$ all signs are positive, so $\bm{z}_{m}=\bm{\Delta}-L\bm{b}_{m}$ , the images sit at $L\bm{b}_{m}$ , and the four colors are $\bm{b}_{m}=\{(0,0),(1,0),(0,1),(1,1)\}$ . Middle. Selector $\hat{m}(\bm{\Delta},\tau=4)$ learned by the min-of-charts NN (seed 0). Horizon-limited states, where all charts tie, are shown in grey. On the distance-limited states, the learned $\hat{m}(\bm{\Delta})$ agrees with the analytic value (colors in left plot) everywhere. Right. The nearest-image distance $d=\text{min}_{m}\,\|\bm{z}_{m}\|$ , equal to $\|\bm{\Delta}\|$ in the fundamental domain. It is continuous, with a concave kink at the ridge, where the index $m^{*}$ jumps, and $\|\nabla d\|=1$ on both sides.

Circle $S^{1}$
Configuration	Best validation loss	RMS error against $V^{*}$	$\\|\nabla V\\|$ at capture	Within $5\%$ of $V^{*}$
Fourier-feature	$1.5\ [0.8,4.6]\times 10^{-4}$	$1.36\ [0.93,1.86]$	$5.00\ [4.99,5.01]$	$44\ [19,60]\%$
soft-periodic	$1.5\ [1.2,1.9]\times 10^{-5}$	$8.7\ [7.4,9.5]\times 10^{-3}$	$5.00\ [5.00,5.01]$	$100\%$
min-of-charts	$2.2\ [1.2,9.0]\times 10^{-6}$	$4.4\ [3.6,6.8]\times 10^{-3}$	$5.00\ [5.00,5.00]$	$100\%$
geometric	$\bm{2.1\ [1.6,2.6]\times 10^{-6}}$	$\bm{4.3\ [3.9,4.8]\times 10^{-3}}$	$5.00\ [5.00,5.00]$	$100\%$
Torus $\mathbb{T}^{2}$

Table 1: Metrics for the four configurations on the three geometries

\mathbb{T}^{n}

(

n=1,2,4

), ten seeds each, with the same training budget within a geometry. (

5\times 10^{5}

iterations on

S^{1}

10^{6}

\mathbb{T}^{2}

and

\mathbb{T}^{4}

). Entries are the median over seeds with the seed range in brackets (in the units of the entry). Best validation loss is Eq. ( 11 ) on a fixed validation set; for the soft-periodic configuration it includes the extra term of Eq. ( 15 ). The RMS is computed against the analytic

V^{*}

2\times 10^{5}

uniform states (

\tau\sim U(0.05,8)

d>\varepsilon

);

\|\nabla V\|

at capture is the mean gradient norm in the band close to the capture set,

d\in(\varepsilon,1.5\varepsilon]

\tau\sim U(0.5,7.5)

, whose analytic value is

5

; the last column is the share of the same

2\times 10^{5}

states within

5\%

relative error of

V^{*}

(with the caveat in the item 5 at the beginning of the section). Bold marks the best median within each geometry (lowest loss and error, highest share within

5\%

), with ties all marked.

Figure 3: Validation loss on

S^{1}

\mathbb{T}^{2}

and

\mathbb{T}^{4}

(left to right) for ten seeds per configuration, computed from Eq. ( 11 ) evaluated on a fixed random sampling. The Fourier-feature NN (red) ends

70

–

180

times above the min-of-charts (orange) and geometric (blue) NNs, which descend together. The soft-periodic NN (green), whose loss includes the additional term ( 15 ), ends between the two groups on

S^{1}

and

\mathbb{T}^{2}

and above the Fourier-feature NN on

\mathbb{T}^{4}

. All losses are still decreasing at the end of the budget.

Figure 4: Value function (left) and absolute error (right) of every configuration, against the analytic

V^{*}

(gray), evaluated at

\tau=3

of an example seed (seed 0). We show

S^{1}

(top), a cut along the diagonal of

\mathbb{T}^{2}

(middle) and a cut along the main diagonal of

\mathbb{T}^{4}

(bottom). The dotted line is the cone

d_{c}=\varepsilon+\Delta v\,\tau

, beyond which

V^{*}=\tau

Figure 5: Value function

V

averaged over directions: at each geodesic distance

d

V

is evaluated on all directions of the sphere

\|\bm{\Delta}\|=d

inside the fundamental domain (

2

S^{1}

720

\mathbb{T}^{2}

512

\mathbb{T}^{4}

), and the mean and standard deviation are computed. The solid vertical line marks

d=1

, beyond which only part of the sphere lies inside the domain. The dotted vertical line marks the cone,

d_{c}

. Same run and panel disposition as Fig. 4 . Left: mean over directions with a

\pm 1\sigma

band. The analytic

V^{*}

depends on

d

alone (not on direction), so any spread in the

1\sigma

band is error due to angular variation of the learned solutions. Right : mean absolute error with its

\pm 1\sigma

band.

Figure 6: Structure maps on $\mathbb{T}^{2}$ at $\tau=4.7$ (seed 0): from left to right columns, learned value $V_{\text{NN}}$ , gradient norm $\|\nabla_{\bm{\Delta}}V\|$ , pointwise residual $|r|$ from Eq. ( 10 ) (log scale) and absolute error $|V_{\text{NN}}-V^{*}|$ , as a function of the wrapped coordinate $\bm{\Delta}$ . In each row, we show the four configurations. The dashed red circle is the cone $d_{c}$ . The Fourier-feature NN has a gradient $\|\nabla_{\bm{\Delta}}V\|$ of correct magnitude $1/\Delta v$ almost everywhere, but a lattice of folds instead of a cone in the value $V$ , with the residual concentrated on the fold lines and an error that grows from the capture set outward reaching $O(1)$ values. The soft-periodic NN has the correct shape of the value $V$ , with maximum absolute error of $10^{-1}$ concentrated around the cone. The min-of-charts and geometric NNs are within $3\cdot 10^{-2}$ of $V^{*}$ everywhere.

Figure 7: Equivalent structure maps as in Fig. 6 , but for $\mathbb{T}^{4}$ , on the slice $\Delta_{3}=\Delta_{4}=0$ at $\tau=4.7$ (seed 0). The Fourier-feature NN shows the same fold lattice as Fig. 6 , with again $O(1)$ absolute errors over most of the slice, increasing as one moves away from the capture set. The soft-periodic NN has the right overall shape, but errors between 0.5 and 1, largest around the cone, especially along the line $\Delta_{1}=0$ and, more weakly, along part of the line $\Delta_{2}=0$ . The min-of-charts and geometric NNs are within about $5\cdot 10^{-2}$ of $V^{*}$ on the whole slice, with the largest errors along the cone, and much lower values inside. In particular, notice that the first two configurations lack to recover the angular symmetry.

Figure 8: Evolution of Fourier-feature NN into almost-everywhere, folded solutions. The value function

V(\Delta,\tau=3)

of the Fourier-feature NN on

S^{1}

(seed 0) is shown as a red line at different stages of a

10^{5}

–iteration run, from the initialization. The analytic

V^{*}

is shown as a gray line. The gray band is the capture set

\|\Delta\|\leq\varepsilon

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

circle $S^{1}$
candidate ( $\tau=4$ )	value RMS	flagged %	max ratio $\hat{\lambda}_{\max}/C^{*}$
$V^{*}$ (analytic)	—	$0.0$	$0.0$
planted sawtooth	—	$8.2$	$100$
Fourier-feature	$1.36\ [0.93,1.86]$	$11.6\ [5.2,18.8]$	$212\ [88,568]$
soft-periodic	$8.7\ [7.4,9.5]\times 10^{-3}$	$\bm{0.0\ [0.0,0.0]}$	$\bm{0.8\ [0.4,1.1]}$
min-of-charts	$4.4\ [3.6,6.8]\times 10^{-3}$	$\bm{0.0\ [0.0,3.6]}$	$1.8\ [0.4,5.9]$

Appendix

Table 2: Monte Carlo semiconcavity test at

\tau=4

(

4\times 10^{4}

probes; eight random directions each and one on

S^{1}

;

h=0.02

; the region

d\leq 3\varepsilon

and concave kinks are masked; a probe is flagged when

\hat{\lambda}_{\max}>2\,C^{*}

), applied to

V^{*}

, to a planted sawtooth (a

\tau

-independent triangle wave in the first coordinate

\Delta_{x}

, of period

0.5

and amplitude

1.25

, so with slope of magnitude

1/\Delta v

on its flanks and convex kinks at its valleys, which shows what a clearly folded function scores) and to all

120

trained networks; median and range over the ten seeds. The value RMS column repeats Table 1 (

\tau\in(0.05,8)

, not only

\tau=4

). On

S^{1}

the curvature of

V^{*}

vanishes away from its kinks, so its ratio is

0

rather than

1

. Bold marks the lowest median among the trained networks within each geometry, with ties all marked.

Figure 9: Seed census of the

120

runs; one marker per run, colour by configuration, marker shape by topology. Left. Value RMS against

V^{*}

(Table 1 ) grouped by topology. Right. The semiconcavity certificate

\max\,\hat{\lambda}_{\max}/C^{*}

(Table 2 ) against the same RMS; the dashed line is the ratio

10

. Folded and relabelled runs lie above it and correct runs below it, while the

\mathbb{T}^{\,4}

soft-periodic failures, smooth creases rather than folds, straddle it.

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Many core problems in nonlinear systems analysis and control can be recast as solving partial differential equations (PDEs) such as Lyapunov and Hamilton-Jacobi-Bellman (HJB) equations. Physics-informed neural networks (PINNs) have emerged as a promising mesh-free approach for approximating their solutions, but in most existing works there is no rigorous guarantee that a small PDE residual implies a small solution error. This paper develops verifiable error bounds for approximate solutions of Lyapunov and HJB equations, with particular emphasis on PINN-based approximations. For both the Lyapunov and HJB PDEs, we show that a verifiable residual bound yields relative error bounds with respect to the true solutions as well as computable a posteriori estimates in terms of the approximate solutions. For the HJB equation, this also yields certified upper and lower bounds on the optimal value function on compact sublevel sets and quantifies the optimality gap of the induced feedback policy. We further show that one-sided residual bounds already imply that the approximation itself defines a valid Lyapunov or control Lyapunov function. We illustrate the results with numerical examples.

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