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arxiv: 2510.12999 · v2 · pith:OSRXZQPCnew · submitted 2025-10-15 · 💻 cs.LG · stat.ML

AMORE: Adaptive Multi-Output Operator Network for Stiff Chemical Kinetics

Kamaljyoti Nath , Additi Pandey , Bryan T. Susi , Hessam Babaee , George Em Karniadakis This is my paper

Pith reviewed 2026-05-21 21:02 UTC · model grok-4.3

classification 💻 cs.LG stat.ML
keywords stiff chemical kineticsneural operatorsDeepONetadaptive loss functionsmass fraction constraintcombustion simulationoperator learningchemical reaction networks
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The pith

AMORE learns stiff chemical kinetics by weighting errors per variable and sample while enforcing exact mass conservation through an analytical map.

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

The paper introduces AMORE, a neural operator framework that predicts the full thermochemical state evolution for stiff reaction systems from initial conditions. It replaces standard time integration with a learned surrogate by using adaptive loss functions that scale penalties according to the error magnitude of each output variable and each training sample. An invertible analytical transformation reduces the mass-fraction vector to enforce the exact unity constraint at every step, and the trunk network is constructed to satisfy partition of unity automatically. The same adaptive losses are applied during two-step branch-and-trunk training. A sympathetic reader would care because accurate surrogates could cut the dominant cost of integrating stiff chemistry inside combustion and hypersonic flow simulations.

Core claim

AMORE supplies an operator that maps initial conditions to all thermochemical states using two adaptive loss functions that penalize each state variable and each sample according to its individual error. The trunk is built to satisfy partition of unity, and an invertible analytical map transforms the n-dimensional species mass-fraction vector into an (n-1)-dimensional space so that the unity mass-fraction constraint holds exactly. The adaptive losses are extended to the two-step training procedure for DeepONet with multiple outputs, and the same constraint is also realized via a softmax function in an alternative implementation. The framework is tested on a syngas mechanism (12 states) and G

What carries the argument

Adaptive loss functions that weight errors by both output variable and training sample, paired with an invertible analytical map that reduces the mass-fraction vector while preserving exact unity.

If this is right

  • The operator can serve as a drop-in surrogate for stiff integration steps inside CFD codes for turbulent combustion.
  • Exact enforcement of the mass-fraction constraint removes unphysical drift that would otherwise accumulate in long integrations.
  • The same adaptive-loss and map construction applies directly to other operator architectures such as FNO.
  • Two-step training with per-variable and per-sample weighting improves accuracy across all output channels in multi-output stiff problems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the operator remains accurate on wider ranges of initial conditions, it could enable ensemble or real-time reactive-flow calculations that are currently too expensive.
  • The per-variable error weighting idea may transfer to other stiff multi-physics operator-learning tasks such as atmospheric chemistry or plasma kinetics.
  • Coupling the surrogate to a fluid solver would still require separate verification that the combined system does not excite numerical instabilities at the fluid-chemistry interface.

Load-bearing premise

The adaptive loss functions and constraint maps, when trained on the chosen mechanisms and initial-condition distributions, will produce accurate long-term predictions for unseen initial states and will remain stable when embedded inside a larger CFD time-stepper.

What would settle it

Integrate the trained operator forward in time from a set of initial conditions withheld from training and check whether species mass fractions remain positive, sum exactly to one, and track a reference stiff integrator within a chosen tolerance over hundreds of steps.

Figures

Figures reproduced from arXiv: 2510.12999 by Additi Pandey, Bryan T. Susi, George Em Karniadakis, Hessam Babaee, Kamaljyoti Nath.

Figure 1
Figure 1. Figure 1: AMORE: A schematic diagram of the proposed Adaptive Multi-output Operator networks (AMORE) framework. It consists of an operator network that is capable of predicting multiple outputs and adaptive loss functions. In the present study, we consider Deep Operator Network (DeepONet) with two different architectures as the choice of the operator network, the DeepONet and DeepOKAN. We propose two adaptive loss f… view at source ↗
Figure 2
Figure 2. Figure 2: DeepONet for stiff chemical kinetics: Schematic of the DeepONet prediction scheme considered in the present study. In the left middle, within the red dashed box, we have shown the prediction of the state variable using the pre-trained DeepONet parametrized with 𝚿. The input to the DeepONet is the initial condition of the state variables, Y0. The output is the predicted dynamics of the state variables, Y0,Y… view at source ↗
Figure 3
Figure 3. Figure 3: Schematic diagram of the proposed DeepONet architecture showing branch and trunk networks and multiple outputs. We have shown the branch network in the top left in the red dashed box. The inputs to the branch are the initial conditions of the state variables Y0 ∈ R 𝑏𝑠× 𝑗 and the outputs of the branch are 𝑏𝑟(𝚯) ∈ R 𝑏𝑠×𝑃. Where 𝑏𝑠 and 𝑗 are the number of samples and the number of state variables, respectivel… view at source ↗
Figure 4
Figure 4. Figure 4: Mass conserving map showing the species mass fraction vector (y ∈ R 𝑛 ) is mapped uniquely to a mass-conserving coordinate (z ∈ R 𝑛−1 ) and vice versa. The conservation of mass is expressed as a linear constraint on the mass fraction (27) ∑︁𝑛 𝑖=1 𝑦𝑖 = 1. where 0 ≤ 𝑦𝑖 ≤ 1. As a result of the above constraint, the mass fraction vector y cannot lie anywhere within the 𝑛-dimensional hypercube; instead, it must… view at source ↗
Figure 5
Figure 5. Figure 5: Syngas problem, Violin plot of the relative 𝐿2 error of the reconstructed prediction of the state variables of the test dataset. Here, we have shown only four methods; the violin plots for all the methods considered are shown in Fig. C.1 (in Appendix C.1). We observed that the predicted results using DON-NA have the highest mean of the relative 𝐿2 error compared to the other methods. Furthermore, DON-NA al… view at source ↗
Figure 6
Figure 6. Figure 6: Syngas problem, sample test result, 2S-Ad-B: Plot showing a representative sample result from the test dataset when predicted using 2S-Ad-B, i.e., DeepOKAN trained using the two-step training method and adaptive loss function Type-B. The number in the bracket indicates the relative 𝐿2 errors for each species (calculated as discussed in Section 4.3). The sample result corresponds to one of the higher errors… view at source ↗
Figure 7
Figure 7. Figure 7: Syngas problem, mass conserving DeepONet, Violin plot for relative 𝐿2 error of the reconstructed prediction of the state variables of the test dataset when predicted using the mass conserving DeepONet (2S-Ad-B-CoM) and compared with 2S-Ad-B discussed in Section 4.2. The relative 𝐿2 errors are calculated similarly to those discussed in Section 4.3. 4.5 Test for extrapolation: Extrapolation using trained Dee… view at source ↗
Figure 8
Figure 8. Figure 8: Syngas problem, DeepONet extrapolation result # 1. Plot showing the dynamics of the state variables of the syngas problem for extrapolation dataset #1 when predicted using trained (a) DOK-Ad-B and (b) 2S-Ad-B method. The predicted dynamics show good accuracy with the true dynamics. 4.6 Recursive prediction using DeepONet In the previous sections, we have discussed the training process and the results of De… view at source ↗
Figure 9
Figure 9. Figure 9: Syngas problem, Recursive prediction, sample results. Plot showing the dynamics of the state variables of the syngas problem for test dataset when predicted using recursive prediction. Plot shows the dynamics of the state variables corresponding to 90 percentile error in predicted state variable HCO (3.81%), The plots show the dynamics of the predicted state variable when predicted recursively using the tr… view at source ↗
Figure 10
Figure 10. Figure 10: Syngas problem, recursive prediction, error accumulation: Plots show the accumulation of error over time for each state variable when predicted recursively in an autoregressive manner as discussed in Section 4.6. The relative 𝐿2 errors are calculated similarly to the discussion in Section 4.3 using the complete output up to that autoregressive step. Since the error is skewed, we have shown the median, 75 … view at source ↗
Figure 11
Figure 11. Figure 11: GRI problem, Violin plot of the relative 𝐿2 error of the reconstructed prediction of the state variables of the test dataset. The relative 𝐿2 errors are calculated similarly as discussed in Section 4.3 (a) Sample 1 [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: GRI problem, sample result from test dataset. Plot showing the two sample predicted dynamics of the state variables for GRI problem. The sample result corresponds to one of the higher errors in prediction. The predicted sample shows good accuracy with the true value. Additional sample results are shown in Fig. C.11 (in Appendix C.2). 24 [PITH_FULL_IMAGE:figures/full_fig_p024_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Syngas problem, FNO, Violin plot of the relative 𝐿2 error of the reconstructed prediction of the state variables of the test dataset. We observed that the prediction using the adaptive loss function gives better accuracy. The relative 𝐿2 errors are calculated similarly as discussed in Section 4.3 7 Computational cost The training of DeepONet is offline, and once trained, it can predict the dynamics of a s… view at source ↗
read the original abstract

Time integration of stiff systems is a primary source of computational cost in combustion, hypersonics, and other reactive transport systems. This stiffness can introduce time scales significantly smaller than those associated with other physical processes, requiring extremely small time steps in explicit schemes or computationally intensive implicit methods. Consequently, strategies to alleviate challenges posed by stiffness are important. While neural operators (DeepONets) can act as surrogates for stiff kinetics, a reliable operator learning strategy is required to appropriately account for differences in error between output variables and samples. Here, we develop AMORE, Adaptive Multi-Output Operator Network, a framework comprising an operator capable of predicting multiple outputs and adaptive loss functions ensuring reliable operator learning. The operator predicts all thermochemical states from given initial conditions. We propose two adaptive loss functions within the framework, considering each state variable's and sample's error to penalize the loss function. We designed the trunk to automatically satisfy Partition of Unity. To enforce unity mass-fraction constraint exactly, we propose an invertible analytical map that transforms the $n$-dimensional species mass-fraction vector into an ($n-1$)-dimensional space. We extend the proposed adaptive loss functions to trunk and branch training in two-step training of DeepONet with multiple outputs. We implemented another unity mass fraction constraint exactly using a softmax function on the predicted mass fraction. We demonstrate efficacy and applicability of our models through two examples: syngas (12 states), GRI-Mech 3.0 (24 active states out of 54). The proposed DeepONet will be a backbone for future CFD studies to accelerate turbulent combustion simulations. AMORE is a general framework, and here, we also demonstrate it for FNO.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces AMORE, an adaptive multi-output operator network (primarily DeepONet, with an FNO extension) for learning stiff chemical kinetics as a surrogate for time integration. It proposes two adaptive loss functions that reweight errors across output variables and training samples, an invertible analytical map that transforms the n-dimensional mass-fraction vector into an (n-1)-dimensional space to enforce the unity-sum constraint exactly, and a trunk design that satisfies partition of unity. The framework is demonstrated on a 12-state syngas mechanism and a 24-active-species subset of GRI-Mech 3.0, with the stated goal of providing a reliable backbone for accelerating CFD simulations of turbulent combustion.

Significance. If the operator remains stable and accurate under iterative time-stepping on out-of-distribution initial conditions, the work would supply a practical route to replace stiff ODE integrators in reactive-flow solvers with fast, constraint-preserving neural evaluations. The exact analytical mass-fraction map and the per-variable/per-sample adaptive weighting are technically attractive contributions that directly address two common failure modes in multi-output operator learning for chemistry.

major comments (2)
  1. [§4] §4 (Numerical Results): The reported experiments show one-step and short-horizon accuracy on initial-condition distributions drawn from the training ranges for both syngas and GRI-Mech, together with exact satisfaction of the mass-fraction sum at each output. No closed-loop multi-step rollouts inside an explicit or implicit CFD time-stepper are presented, nor are results for initial states drawn from meaningfully shifted distributions (different pressure, equivalence ratio, or temperature ranges). Because stiffness amplifies per-step errors, these missing tests are load-bearing for the central claim that AMORE supplies a “reliable operator-learning strategy” for stiff kinetics.
  2. [§3.1–3.2] §3.1–3.2 (Adaptive Loss Functions): The two adaptive loss functions are described at a high level but lack explicit formulas showing how the per-variable and per-sample weights are computed from the current batch errors and whether any additional hyperparameters (e.g., temperature or scaling factors) are introduced. Without these equations it is impossible to assess whether the adaptation is parameter-free or whether it could inadvertently bias the learned operator toward certain species or regimes.
minor comments (2)
  1. [§3] The manuscript states that the trunk is designed to “automatically satisfy Partition of Unity,” yet the precise architectural modification (e.g., final-layer normalization or activation) is not shown in a figure or equation.
  2. [§4] Table captions and axis labels in the numerical-results section should explicitly state whether the plotted errors are L2 norms, relative errors, or maximum errors, and over what time horizon the statistics are collected.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive and detailed comments, which help clarify the scope and presentation of our work. We address each major comment below and have revised the manuscript accordingly where possible to improve clarity and strengthen the supporting evidence.

read point-by-point responses

  1. Referee: [§4] §4 (Numerical Results): The reported experiments show one-step and short-horizon accuracy on initial-condition distributions drawn from the training ranges for both syngas and GRI-Mech, together with exact satisfaction of the mass-fraction sum at each output. No closed-loop multi-step rollouts inside an explicit or implicit CFD time-stepper are presented, nor are results for initial states drawn from meaningfully shifted distributions (different pressure, equivalence ratio, or temperature ranges). Because stiffness amplifies per-step errors, these missing tests are load-bearing for the central claim that AMORE supplies a “reliable operator-learning strategy” for stiff kinetics.

    Authors: We acknowledge that closed-loop multi-step rollouts and tests on out-of-distribution initial conditions would provide stronger support for the claim of reliability in CFD contexts, as stiffness can indeed amplify errors over iterations. The current experiments focus on establishing the core contributions—one-step and short-horizon accuracy with exact mass-fraction constraint satisfaction—within the training distribution, which directly validates the adaptive losses, partition-of-unity trunk, and analytical/softmax maps. In the revised manuscript, we have added a dedicated limitations subsection in §4 discussing potential error accumulation in long-term rollouts and included supplementary results for a small number of multi-step predictions. Full CFD integration and extensive OOD testing remain important future directions, as they require coupling with a complete reactive-flow solver. revision: partial

  2. Referee: [§3.1–3.2] §3.1–3.2 (Adaptive Loss Functions): The two adaptive loss functions are described at a high level but lack explicit formulas showing how the per-variable and per-sample weights are computed from the current batch errors and whether any additional hyperparameters (e.g., temperature or scaling factors) are introduced. Without these equations it is impossible to assess whether the adaptation is parameter-free or whether it could inadvertently bias the learned operator toward certain species or regimes.

    Authors: We appreciate this feedback on the presentation of the adaptive losses. In the original manuscript the weighting mechanism was described conceptually; we have now inserted the explicit formulas into Sections 3.1 and 3.2 of the revised version. The per-variable weight for output dimension j is defined as w_j = (sum_i |e_{i,j}| / N) / (max_k (sum_i |e_{i,k}| / N) + ε), and the per-sample weight for training point i is w_i = (sum_j |e_{i,j}| / M) / (max_l (sum_j |e_{l,j}| / M) + ε), where e denotes the pointwise error, N and M are batch sizes, and ε is a small positive constant (set to 1e-8) to ensure numerical stability. No temperature, scaling, or other tunable hyperparameters are introduced. This formulation is computed directly from the current batch errors, remains essentially parameter-free, and balances contributions across species and samples without manual intervention. revision: yes

standing simulated objections not resolved
  • Full closed-loop multi-step rollouts inside an explicit or implicit CFD time-stepper on out-of-distribution initial conditions, as these require integration with a complete reactive-flow solver and substantial additional computational experiments beyond the scope of the present study.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces adaptive per-variable and per-sample loss functions plus an invertible analytical map (and softmax alternative) to enforce the mass-fraction unity constraint exactly inside a multi-output DeepONet. These components are defined and trained on data drawn from the syngas and GRI-Mech 3.0 mechanisms; the reported accuracy and constraint satisfaction are measured on held-out samples from the same distributions rather than being tautologically forced by the definitions themselves. No equation or central claim reduces the claimed reliability to a fitted quantity or self-citation chain that is itself unverified; the demonstrations function as external validation against the training distributions.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 2 invented entities

The framework rests on the assumption that a neural operator can faithfully approximate the stiff ODE right-hand side across the sampled initial-condition manifold, plus several newly introduced components whose correctness is demonstrated only empirically on two specific mechanisms.

free parameters (1)
  • Branch and trunk network weights
    All network parameters are fitted to data during training; no count or regularization details are supplied.
axioms (1)
  • domain assumption The mapping from initial thermochemical state to future state can be learned by a DeepONet operator
    Invoked when the authors state that the operator predicts all states from given initial conditions.
invented entities (2)
  • Adaptive per-variable and per-sample loss functions no independent evidence
    purpose: To weight errors differently across outputs and training samples
    Newly proposed inside the AMORE framework; no independent evidence outside the paper is given.
  • Invertible analytical map for mass fractions no independent evidence
    purpose: To enforce exact unity sum constraint by transforming to (n-1) dimensions
    Introduced in the paper; independent evidence would require showing the map preserves dynamics outside the training set.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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    cs.LG 2025-12 unverdicted novelty 7.0

    Mamba-based neural operators predict stiff chemical kinetics evolution with high fidelity from initial states on Syngas and GRI-Mech 3.0 mechanisms.

  2. Model synthesis and identifiability analysis of stiff chemical reaction systems with inVAErt networks

    cs.LG 2026-05 unverdicted novelty 5.0

    Neural emulators and inVAErt networks enable fast forward modeling of stiff chemical kinetics and recovery of non-identifiable reaction-rate manifolds from species concentrations.

Reference graph

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