Exact and approximate error bounds for physics-informed neural networks

Augusto T. Chantada; Claudia G. Sc\'occola; Luca Gomez Bachar; Pavlos Protopapas; Susana J. Landau

REVIEW 3 major objections 5 minor 1 cited by

Exact and approximate error bounds for physics-informed neural networks

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper proves residual-only, arbitrarily tight a posteriori error bounds for PINN solutions of nonlinear first-order ODEs, exact for the Riccati case.

desk verdict Worth engaging: the residual-based exact bound for Riccati PINN errors is new and useful, but the exactness claim needs a convergence proof and the examples need R/K reporting. read the letter →

arxiv 2411.13848 v1 pith:COL2OX2N submitted 2024-11-21 cs.LG cs.NAmath.NA

Augusto T. Chantada , Pavlos Protopapas , Luca Gomez Bachar , Susana J. Landau , Claudia G. Scóccola This is my paper · ORCID

classification cs.LGcs.NAmath.NA MSC 65L7065L0568T07

keywords physics-informedneuralnetworksaposteriorierrorboundsfirst-orderODEsRiccatiequationresidual-basedTaylorseriesexpansionnetworksolvers

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks whether a trained physics-informed neural network (PINN) can certify its own error for a nonlinear first-order ODE using only quantities already available after training: the network's output and its residual. It establishes a general series representation of the error as a sum of recursively defined terms built from the Taylor coefficients of the equation's nonlinearity, and from that series derives two bounds. For a general smooth nonlinearity it gives an approximate bound by truncating the series with tolerance-controlled stopping criteria. For the Riccati equation, where the nonlinearity is quadratic, it proves an exact bound that is also arbitrarily tight as the truncation order grows. The practical payoff is that a user can report a rigorous error ceiling without ever calling a numerical ODE solver.

What carries the argument

The carrying object is the recursive error decomposition $\eta_j$, obtained by substituting $u = v + \eta$ into the ODE, Taylor-expanding $f(v+\eta,t)$ in $\eta$, and collecting terms of equal power. With $F_n = \frac{1}{n!}\frac{\partial^n f}{\partial v^n}\big|_{\eta=0}$ and $q(t) = \int_{t_0}^t F_1(t')\,dt'$, the zeroth term $\eta_0$ carries the initial mismatch and the residual, and each $\eta_j$ is built from convolutions of lower-order $\eta$'s against the $F_n$'s. For the Riccati case all $F_n$ vanish for $n > 2$, so the recurrence collapses to a quadratic convolution and the tail satisfies $|\eta_j(t)| \le R\,[RK(t-t_0)]^j e^{-q_\downarrow(t)}$ by induction. Summing that geometric tail yields Eq. (14), with $R$ measuring the worst-case effect of the initial mismatch and residual, $K$ the worst-case quadratic coefficient, and $q_\uparrow, q_\downarrow$ separating the growth and decay parts of $F_1$.

What would settle it

Take a smooth nonlinearity such as $f(u,t)=u^3$ with a PINN solution $v(t)$ that has a small residual, compute the first several $\eta_j$ by numerical quadrature, and compare the partial sums with the true error obtained from a high-accuracy RK45 solve; if the partial sums do not approach the true error while the residual stays small, the series representation (5) fails. For the Riccati bound, choose coefficients where $RK(t-t_0)$ crosses $1$ inside the domain and check numerically whether Eq. (14) remains valid; a counterexample would show the condition is load-bearing.

Watch

Extended reading notes

Core claim

The central claim is that for a first-order ODE $\frac{du}{dt} + f(u,t) = 0$ with smooth $f$, the total error $\eta = u - v$ between the true solution and a PINN solution $v$ can be written as $\eta(t) = \sum_{j=0}^\infty \eta_j(t)$, where $\eta_0$ is determined by the initial-condition mismatch and the residual $r(t)$, and each later $\eta_j$ is a recursive integral involving Taylor coefficients $F_n$ of $f$ around $v$. In the quadratic case $f(u,t) = C(t)u^2 + B(t)u + A(t)$, the paper proves $$|\eta(t)| \le \left|\sum_{j=0}^J \eta_j(t)\right| + \frac{R\,[RK(t-t_0)]^{J+1}\,$e^{{-q_\downarrow(t)}}$}{1 - RK(t-t_0)}$$ whenever $RK(t-t_0) < 1$, with $R$ and $K$ computable from the residual, the initial mismatch, the coefficients, and monotone decompositions of the linearized phase $q$. Because the tail term tends to zero as $J$ grows and the first term tends to the true error, the bound is exact in the limit and arbitrarily tight at any finite $J$. The methods use only residual information and equation structure, not a numerical reference solution.

Load-bearing premise

The argument requires the Taylor expansion of $f(v+\eta,t)$ around $\eta=0$ to converge on the whole domain and the recursively defined series sum $\sum \eta_j$ to equal the true error; the paper's stated $C^\infty$ assumption on $f$ does not by itself guarantee either, and the exact bound also needs $RK(t-t_0) < 1$ throughout the domain.

Editorial extensions

If this is right

For Riccati-type first-order ODEs, a trained PINN can be accompanied by a certified, arbitrarily tight error envelope computed only from its residual and the equation's coefficients.
The approximate bound extends the same residual-only certification to any smooth nonlinearity, at the cost of a truncation criterion rather than a rigorous tail estimate.
The order $J$ can be chosen automatically from a user-supplied tolerance via Eq. (15), so the bound adapts to the quality of the trained network.
Because the bound is a posteriori, it becomes sharper as the loss decreases; the examples show low-loss PINNs need only $J=1$ or $J=2$ for useful bounds.
The authors state the extension to higher-order ODEs and systems of ODEs as future work, implying the method is presently limited to scalar first-order equations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A testable extension is to apply the same tail-bounding idea to polynomial nonlinearities of degree $N$, where $F_n$ vanish beyond $N$ and the convolution sum becomes finite, potentially yielding an exact bound analogous to the Riccati one.
The geometric tail argument suggests that any nonlinearity with bounded Taylor coefficients on the relevant domain could yield a computable tail bound, provided $R$ and $K$ are replaced by suitable envelope constants.
One implicit consequence is that the bound could serve as a training monitor: evaluating $RK(t-t_0)$ during training tells whether the exact bound's validity condition can ever be met on the chosen domain.
For systems of ODEs, a componentwise application would require controlling cross-coupling terms, which the scalar proof does not address; this is a natural next test.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The new thing is a residual-based a posteriori bound on the PINN error for nonlinear first-order ODEs, with an exact version for the Riccati case. That is a genuine extension of the linear-ODE and linear-PDE bounds in Liu et al. (2022, 2023), and it is useful: the bound is computed from the residual and equation coefficients, no reference numerical solution needed, and Eq. (14) is arbitrarily tight when it applies. The authors also say plainly that the general approximate bound is not guaranteed, which I appreciate.

For the Riccati case, the derivation is mostly coherent. f is a quadratic polynomial, so the Taylor expansion step is finite and exact; the induction bound on |eta_j| is straightforward and the geometric tail summation under RK(t-t0)<1 is fine. The main gap is not the Taylor step but the identity eta = sum_j eta_j. The paper states it for f in C^infty, which is false for flat smooth f; for Riccati that objection drops out, but then the authors still need to prove the infinite sum is the actual solution: absolute convergence of the eta_j follows from their bound, but they do not show that the series of derivatives converges uniformly, so passing d/dt through the sum is not justified. This is repairable, but until it is written down, 'exact' overstates what is proved.

Second soft spot: R and K are not reported for the examples, so the validity condition RK(t-t0)<1 is not auditable. A small table would fix that. Third, there is no code or training configuration, so the figures cannot be reproduced. Quadrature error from the trapezoidal rule is also not controlled.

If I were the editor, I would send this to review and ask for a repaired convergence argument, R/K reporting, and code. The exact Riccati result is worth having, and the authors are transparent about the limits of the approximate bound. It is not a takedown; it is a promising workshop paper that needs one more round of rigor.

Referee Report

3 major / 5 minor

Summary. This paper proposes a posteriori error bounds for PINN solutions of first-order nonlinear ODEs. The authors start from the residual equation (4) and derive a formal series representation of the error η=u−v, Eq. (5), with terms η_j defined recursively in Eq. (6). For a general C∞ nonlinearity this series is used to construct approximate bounds (Eqs. (8) and (12)) based on truncating the sum. For Riccati equations f(u,t)=C(t)u^2+B(t)u+A(t), they derive a claimed exact bound, Eq. (14), with a geometric tail term controlled by constants R and K, valid when RK(t−t0)<1. Two numerical experiments, one on a logistic-type equation and one on a cosmological Riccati equation, show that the bounds track the true error for progressively better-trained PINNs.

Significance. The main contribution is a computable, arbitrarily tight upper bound on the PINN error for Riccati equations that uses only the residual and equation coefficients, with no constants fitted to the true error; the induction proof of the geometric tail bound is a useful technique. The approximate bounds for general nonlinear first-order ODEs are simple and appear effective in the example, though the authors correctly state that they are not guaranteed. If the convergence of the series representation is established, the Riccati bound would be a solid and novel result in the PINN error-bound literature.

major comments (3)

[Sec. 2 and Appendix A, Eq. (5)] The derivation of Eq. (5) uses the Taylor expansion f(v+η,t)=∑_{n=0}^\infty F_n η^n and then formally identifies η with ∑η_j. Under the stated assumption f∈C^∞ this expansion need not hold: for a flat smooth function such as f(u)=exp(−1/u^2) (with f(0)=0), all derivatives at u=0 vanish, so the machinery produces only η_0 and cannot represent the true error. Therefore Eq. (5) is not a valid general expression for the error as claimed in the abstract. Either strengthen the regularity assumption to real analyticity in u, with uniform convergence on the relevant interval, or explicitly present Eq. (5) as a formal expansion underlying heuristic bounds; the latter is consistent with the caveat in §2.1 that the approximate bounds are not guaranteed.
[Sec. 2.2 and Appendix B, Eq. (14)] The exact Riccati bound is exact only if the series η=∑_{j=0}^\infty η_j is the actual solution of Eq. (4). The paper proves the bound |η_j(t)|≤R[RK(t−t0)]^j e^{−q↓(t)} (Eq. (24)), which gives uniform absolute convergence under RK(t−t0)<1, but it does not prove that the limit satisfies Eq. (4); this requires uniform convergence of the differentiated series or an equivalent stability argument. As written, Eq. (14) bounds the formal series, not necessarily the true PINN error. For the Riccati case f is polynomial, so the Taylor step is exact, but the missing piece is a convergence theorem showing that ∑η_j solves Eq. (4) under the stated condition.
[Sec. 3, Fig. 1b and Eqs. (14)-(15)] The validity condition RK(t−t0)<1 and the tail term in Eq. (14) depend on R and K, but the manuscript does not report the values of R, K, or the maximum of RK(t−t0) over the domain for the example (17). This makes it impossible to verify the condition or reproduce the reported J(ε). Please report these quantities for each trained PINN, or release the evaluation script.

minor comments (5)

[Sec. 2.2, after Eq. (14)] The expression 'lim_{J−→+∞}' appears to be a typo; it should read 'lim_{J→+∞}'.
[Eq. (15)] The expression inside the ceiling can become negative when ε is not chosen sufficiently small, which would yield a negative J despite the stated condition J∈Z≥0; please clarify how J is selected in that regime.
[Eq. (6b)] The notation ∑_{j1+⋯+jj−k=k+1} is difficult to parse; a brief explanation, together with the Riccati specialization where the only nonzero term is k=j−2, would improve readability.
[Sec. 3] The integrals defining the bounds are evaluated with the trapezoidal rule, but no quadrature error estimate is given. Since the Riccati bound is called exact, please state whether the plotted curves are the analytic bounds evaluated numerically and, if so, discuss the numerical error introduced by quadrature.
[Fig. 1] The axes are small and the loss values appear only in the subcaptions; a table listing L, P, J, R, K, and max RK(t−t0) for each trained network would aid reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Eq. (14) and the approximate bounds are computed from the residual, the PINN solution, and equation coefficients, with no fitted constants and no load-bearing self-citation.

full rationale

The paper's claimed derivation is self-contained. Eq. (14) is not fitted to the true error or to the PINN error; R is defined as max_{t*} (|u0-v(t0)| e^{-q_up(t*)} + ∫|r(t')| e^{q_down(t')}dt') and K as max_{t*}|C(t*)|e^{-q_down(t*)}, both computable from the residual r, the PINN solution v, the initial condition, and the Riccati coefficients. The series terms η_j are recursively defined by Eq. (6) from v, r, and derivatives of f, never from u or η. The bound then follows by the triangle inequality and the geometric tail bound (24), which is proved by induction in Appendix B. The self-citations (Liu et al. 2022, 2023; Chantada et al. 2024) are contextual or methodological lineage, and the core bounds are re-derived rather than imported. A rigor gap exists: Eq. (5) assumes convergence of the formal Taylor-series solution under only f∈C^∞, which is not guaranteed for flat smooth f, and the condition RK(t-t0)<1 is stated but R and K are not reported for the examples. These are correctness/reproducibility concerns, not circularity.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The central derivation rests on a formal Taylor expansion of f and on convergence of an infinite recursive series. No constant is fitted to the true error. User-set truncation orders and tolerances affect tightness but not the claimed inequality. No new entities are introduced.

free parameters (3)

J (truncation order in exact bound, Eq. 14)
Chosen by the user or via Eq. (15) from a tolerance epsilon; controls tightness of the bound, not its validity.
P and J (truncation orders in approximate bound, Eq. 12)
Chosen via user tolerances epsilon_abs,P, epsilon_rel,P, epsilon_abs,J, epsilon_rel,J; not fitted to error data.
Tolerances epsilon_abs and epsilon_rel = 1e-6 and 1e-3 for P; 1e-7 and 1e-4 for J in examples
User-set stopping criteria in Eq. (9); they affect which bound is used, not the mathematics.

assumptions (3)

domain assumption The Taylor expansion of f(v+eta,t) around eta=0 represents f on the relevant domain (f is effectively analytic in u).
Sec. 2 states f is C^infinity after Eq. (1), but Eq. (5) and Appendix A require a convergent Taylor expansion for non-polynomial f.
domain assumption The recursive Volterra series sum eta_j converges to the true error eta and can be differentiated termwise.
Appendix A constructs eta_j by formal rearrangement; no convergence proof is given for general f. For Riccati, the RK(t-t0) < 1 bound supplies convergence.
domain assumption The exact solution u exists on I and the PINN solution v is sufficiently differentiable with finite residual r.
Needed for Eqs. (2)-(4); PINN differentiability depends on activation choices that are not stated in Sec. 3.

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Cite this review

Pith. "Pith review of Exact and approximate error bounds for physics-informed neural networks." pith.science (2026). https://pith.science/paper/COL2OX2N

@misc{pith2026241113848,
  author       = {Pith},
  title        = {Pith review of: Exact and approximate error bounds for physics-informed neural networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/COL2OX2N}},
  note         = {Machine review of arXiv:2411.13848}
}

read the original abstract

The use of neural networks to solve differential equations, as an alternative to traditional numerical solvers, has increased recently. However, error bounds for the obtained solutions have only been developed for certain equations. In this work, we report important progress in calculating error bounds of physics-informed neural networks (PINNs) solutions of nonlinear first-order ODEs. We give a general expression that describes the error of the solution that the PINN-based method provides for a nonlinear first-order ODE. In addition, we propose a technique to calculate an approximate bound for the general case and an exact bound for a particular case. The error bounds are computed using only the residual information and the equation structure. We apply the proposed methods to particular cases and show that they can successfully provide error bounds without relying on the numerical solution.

Figures

Figures reproduced from arXiv: 2411.13848 by the authors.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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Reference graph

Works this paper leans on

15 extracted references · 5 canonical work pages · cited by 1 Pith paper

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T. De Ryck and S. Mishra. Error analysis for physics-informed neural networks ( PINNs ) approximating Kolmogorov PDEs . Advances in Computational Mathematics, 48 0 (6): 0 79, Nov 2022. ISSN 1572-9044. doi:10.1007/s10444-022-09985-9. URL https://doi.org/10.1007/s10444-022-09985-9

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

S. Cai, Z. Mao, Z. Wang, M. Yin, and G. E. Karniadakis. Physics-informed neural networks ( PINNs ) for fluid mechanics: a review. Acta Mechanica Sinica, 37 0 (12): 0 1727--1738, Dec 2021. ISSN 1614-3116. doi:10.1007/s10409-021-01148-1. URL https://doi.org/10.1007/s10409-021-01148-1

doi:10.1007/s10409-021-01148-1 2021

[2] [2]

A. T. Chantada, S. J. Landau, P. Protopapas, C. G. Sc\'occola, and C. Garraffo. Faster Bayesian inference with neural network bundles and new results for f( R ) models. Phys. Rev. D, 109: 0 123514, Jun 2024. doi:10.1103/PhysRevD.109.123514. URL https://link.aps.org/doi/10.1103/PhysRevD.109.123514

doi:10.1103/physrevd.109.123514 2024

[3] [3]

F. Chen, D. Sondak, P. Protopapas, M. Mattheakis, S. Liu, D. Agarwal, and M. D. Giovanni. NeuroDiffEq : A Python package for solving differential equations with neural networks. Journal of Open Source Software, 5 0 (46): 0 1931, 2020. doi:10.21105/joss.01931. URL https://doi.org/10.21105/joss.01931

doi:10.21105/joss.01931 1931

[4] [4]

De Felice, S

A. De Felice, S. Mukohyama, and S. Tsujikawa. Density perturbations in general modified gravitational theories. Phys. Rev. D, 82: 0 023524, Jul 2010. doi:10.1103/PhysRevD.82.023524. URL https://link.aps.org/doi/10.1103/PhysRevD.82.023524

work page doi:10.1103/physrevd.82.023524 2010

[5] [5]

De Ryck and S

T. De Ryck and S. Mishra. Error analysis for physics-informed neural networks ( PINNs ) approximating Kolmogorov PDEs . Advances in Computational Mathematics, 48 0 (6): 0 79, Nov 2022. ISSN 1572-9044. doi:10.1007/s10444-022-09985-9. URL https://doi.org/10.1007/s10444-022-09985-9

doi:10.1007/s10444-022-09985-9 2022

[6] [6]

De Ryck, A

T. De Ryck, A. D. Jagtap, and S. Mishra. Error estimates for physics-informed neural networks approximating the Navier–Stokes equations . IMA Journal of Numerical Analysis, 44 0 (1): 0 83--119, 01 2023. ISSN 0272-4979. doi:10.1093/imanum/drac085. URL https://doi.org/10.1093/imanum/drac085

work page doi:10.1093/imanum/drac085 2023

[7] [7]

H. Jin, M. Mattheakis, and P. Protopapas. Physics-informed neural networks for quantum eigenvalue problems. In 2022 International Joint Conference on Neural Networks (IJCNN), pages 1--8, 2022. doi:10.1109/IJCNN55064.2022.9891944. URL https://doi.org/10.1109/IJCNN55064.2022.9891944

arXiv 2022

[8] [8]

Lagaris, A

I. Lagaris, A. Likas, and D. Fotiadis. Artificial neural networks for solving ordinary and partial differential equations. IEEE Transactions on Neural Networks, 9 0 (5): 0 987--1000, 1998. doi:10.1109/72.712178. URL https://doi.org/10.1109/72.712178

doi:10.1109/72.712178 1998