SALTO: Solver-Anchored Lifted Transfer Operators for Nonlinear Event Inference

• A single classical adjoint is linear in source amplitude, whereas finite-amplitude measurements are nonlinear across a source family.Inference therefore needs sensitivity over a family of sources, not a single vector.Source: §1 · pp. 1–2
• SALTO uses a Koopman lift to turn finite-amplitude source prediction into an inner product with a reusable lifted adjoint.One backward solve per sensor produces the adjoint; each candidate then requires one encoder pass.Source: §1 · p. 2
• Three constraints make SALTO reduce to the simulator’s tangent dynamics and classical adjoint as source amplitude vanishes.The observables are centered to vanish to second order, the state dynamics use the solver’s linearized propagator, and the lifted propagator is block-triangular.Source: §1 · p. 2
• Across three nonlinear PDE systems, SALTO is reported to predict finite-source measurements more accurately than classical models and to infer sources nearly as accurately as solver inversion.The reported contributions also include matching the solver’s joint posterior on reaction-diffusion at one-twentieth its cost and serving new sensor layouts without retraining.Source: §1 · p. 2

• Koopman theory represents nonlinear dynamics through a linear operator on observables, with EDMD fitting a finite-dimensional operator on a prescribed dictionary.Koopman autoencoders learn the dictionary and matrix together; the transpose of an EDMD matrix acts on observable coefficients.Source: §2 · p. 2
• Adjoints compute gradients of measured functionals with one backward solve, while differentiable surrogates provide gradients whose accuracy is rarely validated against the solver.Sobolev training and derivative-informed neural operators supervise derivatives, with tangent solves required during Sobolev training.Source: §2 · p. 3
• Existing source-inference approaches include adjoint-based linear embeddings, their quadratic correction, amortized simulation-based inference, and reconstructions using measurement domains of dependence.The quadratic correction extends the linear embedding to second order.Source: §2 · p. 3

• The state perturbation evolves from a parameterized source under a nonlinear one-step solver map, and measurements are linear functionals of the final state.The solver supplies the linearized propagators along the base trajectory; the base solution is assumed known and reproducible by the solver.Source: §3 · pp. 3–4
• A lifted state augments the physical state with learned observables and evolves under a block-triangular linear propagator.The physical-state block uses the solver’s propagator, while learned blocks couple features to the state and advance the features.Source: §3 · p. 4
• The lifted adjoint is independent of the source state, so one backward solve per sensor can be reused for every candidate source.Its state block is the classical linear adjoint; the nonlinear duality relation adds a finite-amplitude correction to the classical positional embedding.Source: §3 · p. 4
• Scoring a candidate source uses its encoded source and the lifted adjoint, yielding a nonlinear measurement functional without time stepping.The evaluation costs one encoder pass and two inner products.Source: §3 · p. 4
• The posterior evaluates models on the combined field of interacting sources, with source density estimated from Markov chain Monte Carlo samples.The source parameters include position and intensity, and the likelihood uses a noise width that includes model discrepancy.Source: §3 · pp. 4–5

• An unconstrained lift can have 4%–53% rollout error as amplitude vanishes despite matching SALTO’s forward accuracy, motivating structural constraints.The example is Burgers flow about a quiescent base.Source: §4 · p. 5
• Quadratic centering forces learned features to vanish at the base state with zero Jacobian, leaving them to represent only second-order and higher effects.Without these conditions, the features can inject a constant or spurious linear term into the response.Source: §4 · p. 5
• Solver anchoring fixes the state row to the exact linearized propagator, while block triangularity makes feature evolution depend on features alone.These choices impose structure rather than fitting the full propagator from snapshot pairs.Source: §4 · pp. 5–6
• Under the three constraints, the lifted state block equals the solver’s tangent model, and the lifted adjoint’s state block equals the classical adjoint.The lifted rollout differs from the tangent rollout by O(ϵ²); the identities are verified numerically to machine precision relative to the supplied propagators.Source: §4 · p. 5
• The learned blocks are fitted by constrained ridge least squares, while encoder parameters are trained through multi-step rollout error and the closed-form inner solve.The learned data explain the residual after solver-provided linear transport, and training gradually lengthens the rollout.Source: §4 · p. 5

5.1 SELF-ADVECTION: BURGERS ABOUT AN EVOLVING BASE FLOW

• At the top of the Burgers training range, the lifted model keeps response error below a few per cent while the linearized solver’s error rises to four fifths for the strongest source.With five sensors and 1% noise, one-shot lifted inversion matches solver inversion in intensity across every band and is at least as accurate in position; the quadratic correction reduces linear-embedding intensity error from 15% to 9% but remains less accurate than the lift.Source: §5.1 · pp. 6–7, Figure 2 · p. 7

5.2 SATURATION: TWO-DIMENSIONAL REACTION–DIFFUSION

• On reaction–diffusion, the lifted functional predicts nine measurements within 2.4% at the training-range maximum and 4% beyond it; at intensity 10^-3, the ablation lift’s error is 0.008% versus 0.06% for the linear embedding.At that small intensity, the unconstrained lift is off by 23%, and the neural surrogate’s error exceeds the measurement itself.Source: §5.2 · p. 7
• On reaction–diffusion, the linear embedding underestimates median intensity by 40%, while the quadratic correction diverges for stronger sources beyond its radius of convergence.The lifted functional recovers position to a median 0.55 cells versus 0.46 for solver inversion, matches its 14% intensity error, and takes 12 seconds per estimate versus 21 for the solver.Source: §5.2 · p. 7
• Across 17 overlapping reaction–diffusion pairs, the lifted posterior’s mean is a median 0.4 solver posterior standard deviations from the solver’s, while each posterior costs 13 seconds versus 295.The central 95% intervals contain the true values in 87% of 102 cases for the lift and 85% for the solver; the eight-channel equivariant lift was trained on generic random fields, not the searched blob family.Source: §5.2 · pp. 7–8, Figure 3 · p. 8

5.3 SHEARED ADVECTION: KOLMOGOROV FLOW

• 7.7% measurement error for the lifted functional, versus 41% for the linear embedding and 23% for the quadratic correction, on Kolmogorov flow at the top of the training range.For 60 test sources, position error matches the solver at 0.41 cells and intensity error is 3.9% versus 2.8%; evaluating 13,824 candidates takes 12 seconds versus 214 for the solver.Source: §5.3 · p. 8, Figure 4 · p. 9
• At intensity 10^-3 on reaction–diffusion, the constrained lift’s measurement error is 0.008% versus 23% for the unconstrained lift, and its weakest-source measurement gradient on Kolmogorov flow is within 0.7% of the solver’s.The table reports that only the anchored, centered lift is exact as amplitude vanishes, and that it is the most accurate learned model throughout on Kolmogorov flow.Source: §5.3 · p. 8, Table 2 · p. 9

5 EXPERIMENTS

• On Burgers about a quiescent base, a Carleman lift built on twelve POD modes is five times less accurate than the learned lift.The experiments compare each system with the solver and two classical models, using lifts with 24 channels on Burgers and four channels on the two-dimensional systems in the main accuracy and cost table.Source: §5 · p. 6, §5.3 · p. 8

5.3 SHEARED ADVECTION: KOLMOGOROV FLOW

• One trained lift serves new sensor layouts and measurement times with a single backward solve, and at 1.5T its errors are 13% on reaction–diffusion and 17% on Kolmogorov flow.At that horizon, the linear embedding reaches up to 196% and 63%, respectively.Source: §5.3 · p. 9

6 DISCUSSION AND OUTLOOK

• SALTO provides a reusable nonlinear sensitivity in the original space that is exact at small amplitude by construction.The paper describes this as a route toward efficient and accurate causal inference.Source: §6 · p. 9
• Open questions include how many lifted channels and which architecture a problem requires, and how the lift converges as it grows.Source: §6 · p. 9
• The paper also asks how far the approach extends to chaotic turbulent flows, which are closer to real applications.Source: §6 · p. 9

AI USE STATEMENT

• AI tools assisted with editing, polishing, and visualization, and the authors reviewed and verified all AI-assisted text.The authors take responsibility for the final manuscript content.Source: AI USE STATEMENT · p. 10

A PROBLEM SETUPS AND PROTOCOL

• The appendix provides the three systems, inference protocol, training, posterior sampler, and translation-equivariant blocks needed to reproduce the results.The systems are ordered as in Section 5 and were chosen for their types of nonlinearity rather than their physics.Source: A PROBLEM SETUPS AND PROTOCOL · p. 12

A.1 THE THREE SYSTEMS

• The systems cover Burgers self-advection, reaction–diffusion saturation, and vortex self-advection in sheared Kolmogorov flow.Burgers uses five sensors and 800 training releases; reaction–diffusion and Kolmogorov flow use a one-or-two-blob source family and nine probes.Source: §A.1 · p. 12

A.2 INFERENCE PROTOCOL, TRAINING AND SAMPLING

• The inference protocol uses noisy sparse measurements and a lattice search over candidate source positions and intensities, with source width fixed at 0.45.Posterior sampling uses 96 differential-evolution MCMC chains, annealed likelihood, and relabeling for exchangeable sources; training and hardware details are also specified.Source: §A.2 · pp. 12–13

A.3 TRANSLATION-EQUIVARIANT LEARNED BLOCKS

• On reaction–diffusion, translation-equivariant blocks reduce feature size by three orders of magnitude and yield forward errors of 0.19% with four channels and 0.05% with sixteen, versus 116% for the linearized model.The symmetry makes learned blocks convolutional and splits fitting into one c × c system per wavenumber.Source: §A.3 · p. 13

B STRUCTURAL EVIDENCE AND INFERENCE RESULTS

• The appendix presents structural evidence followed by inference results for one source, two sources, and reuse across sensor layouts and horizons.Source: B STRUCTURAL EVIDENCE AND INFERENCE… · p. 13

B.1 STRUCTURAL EVIDENCE

• Across all three systems, the constrained lift predicts measurements at intensity 10^-3 more accurately than the linear embedding, while the unconstrained lift errs by tens of percent and median intensity error stays within two percentage points of brute force.The two-dimensional results use two-channel lifts, and the unconstrained lift was not trained on Burgers.Source: Table 3 · p. 13, §B.1 · p. 14
• On Burgers, the lifted model keeps forward error below 2% while the linearized solver’s error reaches four fifths of the response, and its inversion matches brute force in intensity in every band.The one-shot inversion is at least as accurate in position as brute force.Source: §B.1 · p. 14, Table 4 · p. 14
• Only the full anchored-and-centered lift achieves measurement error below 10^-4 at intensity 10^-3 on both reaction–diffusion and Kolmogorov flow; constraints trade some strongest-band reaction–diffusion fit for the exact small-amplitude limit.At finite amplitude, the full lift is most accurate in every Kolmogorov band, while the unconstrained lift fits the two strongest reaction–diffusion bands about twice as closely.Source: §B.1 · p. 15, Table 5 · p. 15, Table 6 · p. 15
• Increasing channels from two to four reduces top-band measurement error from 6.0% to 1.6% on reaction–diffusion and from 12.6% to 7.6% on Kolmogorov flow, while improving inference and preserving the small-amplitude limit.Source: §B.1 · p. 15
• For the weakest sources, the constrained lift gives the most accurate learned-model measurement gradient on both systems; at finite amplitude, gradient rankings follow forward-error rankings.The unconstrained lift is more accurate on reaction–diffusion at finite amplitude, while the constrained lift is more accurate on Kolmogorov flow.Source: §B.1 · p. 15

B.2 SINGLE-SOURCE INFERENCE

• On reaction–diffusion, the full lift’s measurement error falls with source intensity, while the unconstrained lift’s error levels off at small amplitude.The figure also compares six models and shows one probe’s measurement over twice the training horizon.Source: Figure 6 · p. 16
• On Kolmogorov flow, the full lift is the most accurate model in every intensity band, and its measurement error falls with source intensity.The figure uses an injection of intensity 3 for the extended-horizon probe trace.Source: Figure 7 · p. 16
• For the weakest sources, the constrained lift gives the most accurate measurement gradient among the learned models on both systems.The comparison uses median relative Jacobian error over twelve sources per band.Source: Table 8 · p. 17
• The lifted functional evaluates the finite-amplitude sensitivity map with one inner product per source position, rather than one nonlinear solve.The sensitivity-map comparisons cover reaction–diffusion and Kolmogorov flow; the linear embedding misses both amplitude and shape.Source: Figure 8 · p. 17, §B.2 · p. 17
• The lifted model’s position accuracy is statistically indistinguishable from the solver on all three systems, while Kolmogorov-flow intensity error is 0.2 to 2.1 percentage points higher.For intensity, the paired interval includes zero on Burgers and reaction–diffusion.Source: §B.2 · p. 17

B.3 TWO SOURCES

• The study tests separated moderate-strength pairs, separated strong pairs reaching twice the training budget, and overlapping pairs using noisy 4 × 4 probe measurements.It evaluates 30 pairs per family on reaction–diffusion and 24 on Kolmogorov flow, using four-channel dense lifts.Source: §B.3 · pp. 17–18
• For the strong separated reaction–diffusion pair, the lifted functional recovers both sources and intensities as accurately as brute force, while the linear embedding underestimates every intensity.Figure 10 shows the forward measurements and inferred source locations and intensities.Source: Figure 10 · p. 19
• On reaction–diffusion, the lifted functional places both sources within one lattice cell in 80–100% of pairs, versus 33–97% for the linear embedding, and infers intensities within 10% versus 34–49% error.It runs 18 to 26 times faster than brute-force inference; on Kolmogorov flow it resolves both sources more often than the linear embedding in every family despite 43–48% interaction.Source: §B.3 · p. 19, Table 10 · p. 19
• Beyond the training horizon, the lifted rollout’s whole-field error is 3.5% through 2T, compared with 81% for the linearized-response superposition.The lifted rollout reproduces the strong pair’s decay, while the superposition remains twice as strong.Source: Figure 11 · p. 20

B.4 REUSE ACROSS SENSOR LAYOUTS AND HORIZONS

• Across two held-out sensor layouts and a held-out horizon, the lift is more accurate than the direct CNN in every column on both systems.The direct readout stays close to the lift but is refitted for each layout and horizon; the parameter MLP is off by 13–130% for the weakest sources and cannot represent pairs.Source: §B.4 · p. 20, Table 11 · p. 20
• At 1.5T, the lift’s measurement error stays within 13% on reaction–diffusion and 17% on Kolmogorov flow in every intensity band, versus up to 196% and 63% for the linear embedding.The direct readout and CNN have no training data beyond T.Source: §B.4 · p. 20, Table 12 · p. 21
• With 50 rather than 400 training trajectories, the lift is up to four times more accurate than a readout fitted to the same trajectories on reaction–diffusion.This result is reported for the scarce-data comparison in Table 12.Source: §B.4 · p. 20, Table 12 · p. 21
• With 400 trajectories, an anchored and centered CNN without the propagator is 5 to 16 times less accurate than the lift on reaction–diffusion and less accurate in every Kolmogorov-flow band.Anchoring and centering alone therefore do not explain the lift’s reported accuracy.Source: §B.4 · p. 21

C BURGERS ABOUT A QUIESCENT BASE: WHERE THE CONSTRAINTS CAME

• The Burgers appendix study uses a quiescent base, the exact heat propagator, a 1.5 horizon, and test intensities up to 3, beyond the training range.It is explicitly distinguished from the three systems in Section 5.Source: C BURGERS ABOUT A QUIESCENT BASE:… · p. 21
• Despite identical 1.3% validation rollout errors, the unconstrained lift amplifies a backward sensor kernel by 217 in norm, while the anchored, centered lift’s state channel remains at 0.64.The unconstrained model’s state channel alone amplifies by 6.1, whereas the heat adjoint decays.Source: C BURGERS ABOUT A QUIESCENT BASE:… · p. 21
• As amplitude falls, the unconstrained lift’s rollout error stays between 4% and 53% across three decades instead of falling with amplitude.Its observables have norm O(10) at zero and its fitted propagator leaves a bias of 10^-5 per step.Source: C BURGERS ABOUT A QUIESCENT BASE:… · p. 21
• For the anchored, centered lift, the bias is at double-precision machine epsilon, its Jacobian matches the heat propagator to that tolerance, and its nonlinear remainder scales as I_s^2.02.Its rollout error decreases without a floor; centering without anchoring leaves a 3% small-amplitude floor.Source: C BURGERS ABOUT A QUIESCENT BASE:… · p. 22