cs.LGMar 17, 2026

Trajectory-Optimized Time Reparameterization for Learning-Compatible Reduced-Order Modeling of Stiff Dynamical Systems

Authors: Joe Standridge, Daniel Livescu, Paul Cizmas

Organizations: Texas A&M University, College Station, Texas 77840 · Los Alamos National Laboratory, Los Alamos, New Mexico 87545

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Abstract

Stiff dynamical systems present a challenge for machine-learning reduced-order models (ML-ROMs), as explicit time integration becomes unstable in stiff regimes while implicit integration within learning loops is computationally expensive and often degrades training efficiency. Time reparameterization (TR) offers an alternative by transforming the independent variable so that rapid physical-time transients are spread over a stretched-time coordinate, enabling stable explicit integration on uniformly sampled grids. Although several TR strategies have been proposed, their effect on learnability in ML-ROMs remains incompletely understood. This work investigates time reparameterization as a stiffness-mitigation mechanism for neural ODE reduced-order modeling and introduces a trajectory-optimized TR (TOTR) formulation. The proposed approach casts time reparameterization as an optimization problem in arc-length coordinates, in which a traversal-speed profile is selected to penalize acceleration in stretched time. By targeting the smoothness of the training dynamics, this formulation produces reparameterized trajectories that are better conditioned and easier to learn than existing TR methods. TOTR is evaluated on three stiff problems: a parameterized stiff linear system, the van der Pol oscillator, and the HIRES chemical kinetics model. Across all cases, the proposed approach yields smoother reparameterizations and improved physical-time predictions under identical training regimens than other TR approaches. Quantitative results demonstrate loss reductions of one to two orders of magnitude compared to benchmark algorithms. These results highlight that effective stiffness mitigation in ML-ROMs depends critically on the regularity and learnability of the time map itself, and that optimization-based TR provides a robust framework for explicit reduced-order modeling of multiscale dynamical systems.

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 min⁡v∗∫0S(v(s)vs(s)2+κ(s)2 v(s)3) ds,\min_{v^{*}}\int_{0}^{S}\left(v(s)v_{s}(s)^{2}+\kappa(s)^{2}\,v(s)^{3}\right)\,\mathrm{d}s, 
Algorithm 3 Trajectory-Optimized Time Reparameterization
Figure 1: Van der Pol oscillator μ∈[103,104]\mu\in[10^{3},10^{4}] reparameterization by method for an integration tolerance of 10−310^{-3}
Figure 2: Van der Pol oscillator μ∈[103,104]\mu\in[10^{3},10^{4}] reparameterization by method for an integration tolerance of 10−610^{-6}
Figure 3: Time reparameterizations for the stiff linear system, off-reference test case μ=101.05\mu=10^{1.05} .
Figure 4: Trajectories for μ=101.05\mu=10^{1.05} in the reparameterized coordinate, τ\tau .
Figure 5: Time reparameterizations for the stiff linear system, off-reference test case μ=103.95\mu=10^{3.95} .
Figure 6: Early-time zoom (first ten milliseconds) for μ=103.95\mu=10^{3.95} , highlighting resolution of the fastest transient scales.
Figure 7: Trajectories for μ=103.95\mu=10^{3.95} in the reparameterized coordinate, τ\tau
Figure 8: Time curves and their derivatives for μ=103.95\mu=10^{3.95} .
  Solver-DirectedExtrema-BasedTrajectory-Optimized
BlockMetricONOFFONOFFONOFF
τ\tau MSE (nondim)y1y_{1}5.99×10−45.99\times 10^{-4}8.65×10−48.65\times 10^{-4}2.33×10−42.33\times 10^{-4}2.54×10−42.54\times 10^{-4}3.74×10−53.74\times 10^{-5}4.82×10−54.82\times 10^{-5}
y2y_{2}8.51×10−38.51\times 10^{-3}1.23×10−21.23\times 10^{-2}5.95×10−45.95\times 10^{-4}6.92×10−46.92\times 10^{-4}4.00×10−54.00\times 10^{-5}5.72×10−55.72\times 10^{-5}
y3y_{3}4.80×10−34.80\times 10^{-3}6.62×10−36.62\times 10^{-3}2.73×10−42.73\times 10^{-4}3.15×10−43.15\times 10^{-4}1.40×10−41.40\times 10^{-4}1.98×10−41.98\times 10^{-4}
y4y_{4}1.50×10−21.50\times 10^{-2}2.07×10−22.07\times 10^{-2}5.63×10−45.63\times 10^{-4}7.32×10−47.32\times 10^{-4}4.44×10−44.44\times 10^{-4}6.42×10−46.42\times 10^{-4}
y5y_{5}8.87×10−48.87\times 10^{-4}1.37×10−31.37\times 10^{-3}3.67×10−43.67\times 10^{-4}4.12×10−44.12\times 10^{-4}4.38×10−54.38\times 10^{-5}6.03×10−56.03\times 10^{-5}
Table 1: SLS τ\tau -MSE for all methods (On- vs Off-reference).
  Solver-DirectedExtrema-BasedTrajectory-Optimized
BlockMetricONOFFONOFFONOFF
MSIE (dim)y1y_{1}6.70×10−46.70\times 10^{-4}9.46×10−49.46\times 10^{-4}2.65×10−42.65\times 10^{-4}2.96×10−42.96\times 10^{-4}1.07×10−41.07\times 10^{-4}1.42×10−41.42\times 10^{-4}
y2y_{2}8.49×10−48.49\times 10^{-4}1.32×10−31.32\times 10^{-3}3.68×10−53.68\times 10^{-5}3.75×10−53.75\times 10^{-5}5.97×10−65.97\times 10^{-6}8.88×10−68.88\times 10^{-6}
y3y_{3}1.65×10−31.65\times 10^{-3}2.61×10−32.61\times 10^{-3}7.64×10−57.64\times 10^{-5}8.12×10−58.12\times 10^{-5}8.94×10−58.94\times 10^{-5}1.35×10−41.35\times 10^{-4}
y4y_{4}4.00×10−44.00\times 10^{-4}6.23×10−46.23\times 10^{-4}1.54×10−51.54\times 10^{-5}1.78×10−51.78\times 10^{-5}2.47×10−52.47\times 10^{-5}3.83×10−53.83\times 10^{-5}
y5y_{5}3.38×10−43.38\times 10^{-4}6.31×10−46.31\times 10^{-4}1.87×10−41.87\times 10^{-4}1.85×10−41.85\times 10^{-4}1.11×10−51.11\times 10^{-5}1.43×10−51.43\times 10^{-5}
Table 2: SLS MSIE (dimensional) for all methods (On- vs Off-reference).
Figure 9: Van der Pol oscillator, off-reference test case μ=102.01\mu=10^{2.01} .
Figure 10: Van der Pol oscillator, off-reference test case μ=102.01\mu=10^{2.01} , emphasizing behavior near a boundary layer.
Figure 11: Van der Pol oscillator, off-reference test case μ=102.01\mu=10^{2.01} , trajectories in the reparameterized coordinate.
Figure 12: Van der Pol oscillator, off-reference test case μ=103.99\mu=10^{3.99} : comparison of time reparameterizations in the extreme stiffness regime.
Figure 13: Van der Pol oscillator, off-reference test case μ=103.99\mu=10^{3.99} : comparison of time reparameterizations in the extreme stiffness regime, emphasis on boundary layer behaviour.
  Solver-DirectedExtrema-BasedTrajectory-Optimized
BlockMetricONOFFONOFFONOFF
τ\tau MSE (nondim)y1y_{1}7.49×10−27.49\times 10^{-2}9.14×10−29.14\times 10^{-2}2.50×10−32.50\times 10^{-3}2.24×10−32.24\times 10^{-3}5.68×10−35.68\times 10^{-3}6.23×10−36.23\times 10^{-3}
y2y_{2}1.05×10−11.05\times 10^{-1}1.21×10−11.21\times 10^{-1}3.30×10−33.30\times 10^{-3}2.98×10−32.98\times 10^{-3}6.95×10−36.95\times 10^{-3}7.70×10−37.70\times 10^{-3}
Time4.68×10−44.68\times 10^{-4}5.15×10−45.15\times 10^{-4}3.20×10−53.20\times 10^{-5}3.17×10−53.17\times 10^{-5}3.31×10−53.31\times 10^{-5}3.53×10−53.53\times 10^{-5}
State9.00×10−29.00\times 10^{-2}1.06×10−11.06\times 10^{-1}2.90×10−32.90\times 10^{-3}2.61×10−32.61\times 10^{-3}6.32×10−36.32\times 10^{-3}6.96×10−36.96\times 10^{-3}
All6.02×10−26.02\times 10^{-2}7.10×10−27.10\times 10^{-2}1.95×10−31.95\times 10^{-3}1.75×10−31.75\times 10^{-3}4.22×10−34.22\times 10^{-3}4.65×10−34.65\times 10^{-3}
Table 3: Van der Pol τ\tau -MSE (nondimensional) for all methods (On- vs Off-reference).
  Solver-DirectedExtrema-BasedTrajectory-Optimized
BlockMetricONOFFONOFFONOFF
MSIE (dim)y1y_{1}6.14×10−16.14\times 10^{-1}5.17×10−15.17\times 10^{-1}3.44×10−13.44\times 10^{-1}3.82×10−13.82\times 10^{-1}2.04×10−12.04\times 10^{-1}2.21×10−12.21\times 10^{-1}
y2y_{2}1.96×10−31.96\times 10^{-3}1.27×10−31.27\times 10^{-3}5.79×10−45.79\times 10^{-4}5.41×10−45.41\times 10^{-4}5.22×10−45.22\times 10^{-4}1.78×10−31.78\times 10^{-3}
State3.08×10−13.08\times 10^{-1}2.59×10−12.59\times 10^{-1}1.72×10−11.72\times 10^{-1}1.91×10−11.91\times 10^{-1}1.02×10−11.02\times 10^{-1}1.11×10−11.11\times 10^{-1}
Table 4: Van der Pol MSIE (dimensional) for all methods (On- vs Off-reference).
Figure 14: HIRES system, off-reference test case μ=102.025\mu=10^{2.025} .
Figure 15: HIRES system, off-reference test case μ=102.025\mu=10^{2.025} .
Figure 16: HIRES system, off-reference test case μ=102.025\mu=10^{2.025} .
Figure 17: HIRES system, off-reference test case μ=103.975\mu=10^{3.975} .
Figure 18: HIRES system, off-reference test case μ=103.975\mu=10^{3.975} .
Figure 19: HIRES system, off-reference test case μ=103.975\mu=10^{3.975} .
  Solver-DirectedExtrema-BasedTrajectory-Optimized
BlockMetricONOFFONOFFONOFF
τ\tau MSE (nondim)y1y_{1}1.65×10−51.65\times 10^{-5}1.68×10−51.68\times 10^{-5}6.86×10−56.86\times 10^{-5}7.94×10−57.94\times 10^{-5}1.11×10−51.11\times 10^{-5}1.93×10−51.93\times 10^{-5}
y2y_{2}1.00×10−31.00\times 10^{-3}9.87×10−49.87\times 10^{-4}3.30×10−43.30\times 10^{-4}3.79×10−43.79\times 10^{-4}4.52×10−54.52\times 10^{-5}4.83×10−54.83\times 10^{-5}
y3y_{3}1.49×10−31.49\times 10^{-3}1.51×10−31.51\times 10^{-3}4.31×10−44.31\times 10^{-4}4.49×10−44.49\times 10^{-4}9.62×10−59.62\times 10^{-5}1.17×10−41.17\times 10^{-4}
y4y_{4}2.18×10−42.18\times 10^{-4}2.10×10−42.10\times 10^{-4}2.12×10−42.12\times 10^{-4}2.32×10−42.32\times 10^{-4}4.81×10−54.81\times 10^{-5}4.99×10−54.99\times 10^{-5}
y5y_{5}1.43×10−41.43\times 10^{-4}1.84×10−41.84\times 10^{-4}2.95×10−42.95\times 10^{-4}2.18×10−42.18\times 10^{-4}3.70×10−53.70\times 10^{-5}4.29×10−54.29\times 10^{-5}
Table 5: HIRES τ\tau -MSE (nondimensional) for all methods (ON- vs OFF-reference).
  Solver-DirectedExtrema-BasedTrajectory-Optimized
BlockMetricONOFFONOFFONOFF
MSIE (dim)y1y_{1}1.55×10−31.55\times 10^{-3}1.58×10−31.58\times 10^{-3}2.47×10−52.47\times 10^{-5}3.55×10−53.55\times 10^{-5}1.70×10−51.70\times 10^{-5}2.55×10−52.55\times 10^{-5}
y2y_{2}5.54×10−55.54\times 10^{-5}5.54×10−55.54\times 10^{-5}1.80×10−61.80\times 10^{-6}2.38×10−62.38\times 10^{-6}9.74×10−79.74\times 10^{-7}9.43×10−79.43\times 10^{-7}
y3y_{3}3.76×10−63.76\times 10^{-6}3.78×10−63.78\times 10^{-6}1.88×10−71.88\times 10^{-7}1.85×10−71.85\times 10^{-7}1.34×10−81.34\times 10^{-8}1.53×10−81.53\times 10^{-8}
y4y_{4}7.42×10−47.42\times 10^{-4}7.55×10−47.55\times 10^{-4}1.14×10−51.14\times 10^{-5}1.45×10−51.45\times 10^{-5}1.11×10−51.11\times 10^{-5}1.25×10−51.25\times 10^{-5}
y5y_{5}7.69×10−57.69\times 10^{-5}7.89×10−57.89\times 10^{-5}1.99×10−61.99\times 10^{-6}2.56×10−62.56\times 10^{-6}1.02×10−61.02\times 10^{-6}1.29×10−61.29\times 10^{-6}
Table 6: HIRES MSIE (dimensional) for all methods (ON- vs OFF-reference).
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