Physics-conforming Latent Twins

Abstract: Surrogate models are central to scientific machine learning, where they enable fast prediction, simulation, inference, and control for complex physical systems. For time-dependent problems, however, accurate interpolation of training trajectories is not sufficient: reliable surrogates should also respect the conservation laws, invariants, admissibility conditions, and dissipative structures that give those trajectories physical meaning. We introduce Physics-conforming Latent Twins, a framework for learning latent surrogate solution operators whose dynamics satisfy selected physical principles by design. The method builds on the Latent Twin formulation by jointly learning an encoder, a decoder, and a latent flow map between arbitrary time-indexed states, while constraining the latent dynamics to preserve or dissipate prescribed structural quantities. We develop a constraint-transfer viewpoint that connects physical structure in the original state space with compatible constraints in latent space, and prove structure-preservation bounds showing how latent enforcement improves control of physical defects after decoding. We also derive algebraic conditions for latent flow maps that preserve linear and quadratic invariants or enforce dissipative inequalities. Numerical experiments on representative ODE and PDE benchmarks demonstrate improved constraint satisfaction, structural fidelity, and qualitative long-time behavior while maintaining accurate surrogate prediction.

• The paper introduces a latent-flow framework that encodes conservation laws, invariants, admissibility, and dissipation directly into learned surrogate dynamics rather than treating them as post hoc checks.
• Its a priori theory shows that compatible latent constraints eliminate structural-violation growth from latent evolution, leaving error controlled by constraint compatibility and autoencoder accuracy.
• Numerical tests across ODEs and PDEs show improved long-term stability and extrapolation, including near-exact mean and energy preservation, while highlighting limitations for nonlinear decoders and local conservation laws.

Physics-conforming Latent Twins (2606.15053) extends the Latent Twins framework [(2606.15053), chung2026latent] from purely data-driven surrogate solution operators to latent dynamics that satisfy selected physical principles by design. The central premise is that conservation laws, invariants, admissibility conditions, and dissipative structures are organizing principles of the dynamics rather than post hoc diagnostics, and that a learned latent representation can serve as the space in which these principles are encoded and propagated. The paper contributes a constraint-transfer theory, algebraic conditions for structure-preserving latent flows, and numerical demonstrations across conservative, Hamiltonian, and dissipative ODE and PDE benchmarks.

Positioning and formulation

The framework is situated among reduced-order modeling (POD, reduced basis), Koopman methods, SINDy-style system identification, physics-informed neural networks (PINNs), and latent-space dynamics identification (LaSDI variants). The distinguishing feature is the mathematical object learned: rather than a projected differential equation, an identified vector field, or a residual-constrained solution field, the central object is a latent flow map $u(s)\mapsto u(t)$ of the form

$\Psi(t,s,u_s) := d\bigl(m_{s\to t}(e(u_s))\bigr),$

with encoder $e$, decoder $d$, and latent map $m_{s\to t}$ trained jointly. Physical structure is encoded through a structural functional $\mathcal C:\mathcal U\to\mathbb R^k$ whose evolution satisfies either $\frac{d}{dt}\mathcal C(u_t)=0$ (conservation) or $\frac{d}{dt}\mathcal C(u_t)\le 0$ (dissipation/admissibility). The framework builds directly on the Latent Twin approximation theorem of chung2026latent, which gives uniform control of the lifted surrogate on a compact invariant set under assumptions of well-posedness, stable discretization, and expressiveness of the encoder–decoder–latent-map triple.

Constraint-transfer theory

The theoretical contribution proceeds in two tiers. The a posteriori result is a direct corollary of trajectory approximation: if $\mathcal C$ is $L_{\mathcal C}$-Lipschitz and the exact flow satisfies $\Psi(t,s,u_s) := d\bigl(m_{s\to t}(e(u_s))\bigr),$0 componentwise, then the positive-part structural defect of the surrogate is bounded by $\Psi(t,s,u_s) := d\bigl(m_{s\to t}(e(u_s))\bigr),$1 times the full Latent Twin approximation error, which includes discretization, autoencoder, and latent-map errors together with the growth factor $\Psi(t,s,u_s) := d\bigl(m_{s\to t}(e(u_s))\bigr),$2. The paper is explicit that this bound is weak: even exact structure preservation by the latent map would not improve it, since the defect is routed through the entire trajectory error.

The sharper a priori result requires a latent constraint $\Psi(t,s,u_s) := d\bigl(m_{s\to t}(e(u_s))\bigr),$3 that is $\Psi(t,s,u_s) := d\bigl(m_{s\to t}(e(u_s))\bigr),$4-compatible with $\Psi(t,s,u_s) := d\bigl(m_{s\to t}(e(u_s))\bigr),$5 via $\Psi(t,s,u_s) := d\bigl(m_{s\to t}(e(u_s))\bigr),$6, meaning $\Psi(t,s,u_s) := d\bigl(m_{s\to t}(e(u_s))\bigr),$7 on the reachable latent set, together with a $\Psi(t,s,u_s) := d\bigl(m_{s\to t}(e(u_s))\bigr),$8-preserving latent flow (exact invariance for equality constraints, monotone non-increase for inequality constraints). Under these conditions the structural defect satisfies

$\Psi(t,s,u_s) := d\bigl(m_{s\to t}(e(u_s))\bigr),$9

and when $e$0 it reduces to $e$1. This is the paper's strongest claim: with a compatible latent constraint enforced architecturally, the latent evolution contributes no structural violation at all, and no amplification through trajectory error or the exponential growth factor appears in the bound. The practical caveat, which the paper concedes, is that the ideal choice $e$2 is unavailable before training because the decoder is learned, so $e$3 must be prescribed from a tractable family and compatibility becomes an empirical property of the trained model.

Algebraic realizations

For a linear reconstruction map and linear latent flow $e$4, the paper proves exact correspondence results: linear physical functionals pull back to $e$5 with preservation iff $e$6, and quadratic functionals pull back to $e$7 with preservation iff $e$8 and dissipation iff $e$9. For the exponential latent flow $d$0, which preserves the semigroup property, these become generator-level conditions: $d$1 for linear conservation, $d$2 for quadratic invariants, and $d$3 for Lyapunov-type dissipation. Nonlinear latent dynamics can be obtained by conjugation $d$4 with an invertible network $d$5, retaining an analytically controlled inner flow. The paper distinguishes three enforcement levels—architectural (strongest guarantee), operator-regularization, and trajectory-level penalties (most flexible, exact only on sampled data)—and notes that for nonlinear decoders the linear/quadratic results should be read as local design principles near the latent data manifold.

Numerical evidence

The ODE benchmarks progress through four regimes. On a stable linear system with ambient dimension $d$6 and intrinsic dimension $d$7, the physics-conforming model with a softly enforced skew-symmetric generator maintains errors around $d$8 over the horizon, versus roughly $d$9 for DMD, while POD achieves near machine precision ($m_{s\to t}$0) as the optimal projection benchmark in this constructed low-rank setting. On a 20-group SIR system (state dimension 60) with soft enforcement of the linear population-conservation constraint, a nonlinear autoencoder with a linear exponential latent flow of dimension $m_{s\to t}$1 outperforms both POD and DMD at the same reduced dimension, and the nonlinear embedding yields prediction errors roughly two orders of magnitude smaller than the linear-autoencoder variant—indicating that the bottleneck is representation expressiveness, not the linear latent flow.

The undamped pendulum provides the clearest demonstration of hard architectural constraints. The latent generator is parameterized as $m_{s\to t}$2 with $m_{s\to t}$3 skew-symmetric and $m_{s\to t}$4, which conserves the quadratic latent energy exactly by construction ($m_{s\to t}$5). Each Latent Twin model has only 508 trainable parameters versus 12,738 for the PINN baseline. Trained on $m_{s\to t}$6 without access to the governing equation, the physics-conforming model remains stable and accurate on extrapolation to $m_{s\to t}$7, whereas the unconstrained Latent Twin accumulates phase and amplitude error and the PINN, despite using the ODE residual, deteriorates outside its training domain. The paper is careful to frame the PINN comparison as qualitative rather than a data-efficiency benchmark. The dissipative benchmark, Rosenbrock gradient flow, uses a trajectory-level penalty on forward-time latent-norm expansion; violations of nonexpansivity are essentially eliminated after training, and the dissipative model tracks the valley under extrapolation where the unconstrained model drifts.

The PDE experiments use convolutional autoencoders with exponential latent flows on the periodic domain $m_{s\to t}$8. For the heat equation ($m_{s\to t}$9), the generator is decomposed as $\mathcal C:\mathcal U\to\mathbb R^k$0 with $\mathcal C:\mathcal U\to\mathbb R^k$1 and $\mathcal C:\mathcal U\to\mathbb R^k$2, combining a hard-projected linear conservation constraint ($\mathcal C:\mathcal U\to\mathbb R^k$3) with a negative-definite dissipative part. The model preserves the spatial mean to near-machine precision and reproduces monotone $\mathcal C:\mathcal U\to\mathbb R^k$4-energy decay. For the wave equation ($\mathcal C:\mathcal U\to\mathbb R^k$5), a purely skew-symmetric generator $\mathcal C:\mathcal U\to\mathbb R^k$6 conserves the latent quadratic norm up to numerical precision; the predicted Hamiltonian deviates by approximately 2% over the rollout horizon $\mathcal C:\mathcal U\to\mathbb R^k$7, with field-level errors concentrated near sharp, phase-sensitive features. The paper notes this is the more demanding setting since errors are not damped by the dynamics.

Limitations

The paper identifies four limitations. Compatibility between a prescribed $\mathcal C:\mathcal U\to\mathbb R^k$8 and the decoder-induced pullback is not automatic and is difficult to guarantee for highly nonlinear decoders. Hard architectural constraints may reduce expressiveness when the latent constraint class mismatches the physical structure after decoding. The theory addresses only global structural functionals and does not yet guarantee local conservation laws or pointwise admissibility after reconstruction. Finally, the numerical experiments are controlled demonstrations rather than exhaustive benchmarks. The comparison with POD on the linear benchmark is also candid: in a setting constructed to admit an optimal low-rank projection, POD remains essentially exact, so the framework's advantage is confined to settings where the representation must be learned jointly with the dynamics.

Conclusion

Physics-conforming Latent Twins formalize the idea that the latent space of a surrogate solution operator can be the locus of physical structure rather than mere compression. The a priori structure-preserving bound—structural defect controlled by compatibility and autoencoder error alone—provides the theoretical justification for architectural enforcement in latent space, and the ODE/PDE experiments show the resulting gains in long-time stability and extrapolation. The open questions left by the paper are specific: learning or adapting the latent constraint itself rather than prescribing it, extending to parameterized operator-learning settings, and moving from global invariants to local, spatially resolved balance laws.