Learning PDE Dynamics between Submanifolds Using Green's Observation Operators

1 INTRODUCTION

• Lower-dimensional inputs and observations remain coupled through the full medium, so the restricted state generally cannot be modeled independently of the surrounding domain.Examples include laser heating observed by infrared cameras and ground emissions measured on a sensor plane.Source: §1 · pp. 1–2
• Classical PDE solvers resolve the entire volume for each new source even when only the response on the observation submanifold is needed.This added computation can limit online prediction or planning.Source: §1 · p. 2
• GObO learns a time-dependent Green’s kernel between source and observation submanifolds, conditioned on the ambient medium, so new sources need not be observed during training.The kernel model is reused across different source terms.Source: §1 · p. 2
• GObO achieves 4–8× lower error under source generalization and over 100× faster evaluation of new sources than black-box neural operators.It also achieves up to 3.5× lower error than the learned Green’s function baseline.Source: §1 · p. 2

2.1 SUBMANIFOLDS OF CLOSED SYSTEMS ARE NOT CLOSED

• Information can travel through the unobserved domain and later influence the observation trace, so the trace alone does not determine its future.The full ambient state determines future evolution; the surrounding volume therefore acts as hidden memory.Source: §2.1 · p. 3

2.2 PDE SOLVERS REQUIRE CLOSED SYSTEMS

• On the heat benchmark, the finite-volume solver is about 2750× slower than GObO because each new source requires another full ambient solve.Table 1 concerns repeated volumetric solves when only a lower-dimensional trace is needed.Source: §2.2 · p. 3, Table 1 · p. 3

2.3 BLACK-BOX APPROACHES MISS KEY OPERATOR STRUCTURE OR DYNAMICS

• Black-box surrogates jointly process the medium and source, requiring another network evaluation for each source without built-in generalization to unseen source configurations.Transient neural Green’s models also rely on a one-step kernel rollout requiring a closed state, which the observed trace does not provide.Source: §2.3 · p. 3

2.4 PROPOSED GOBO DIRECTLY MODELS NON-CLOSED SUBMANIFOLD DYNAMICS

• GObO represents ambient dynamics with a time-dependent kernel between the submanifolds, avoiding volumetric reconstruction and additional network passes for new sources.The structured parametrization also enables recurrent computation that compresses system memory into a running state.Source: §2.4 · p. 3

3 A PRIMER ON GREEN’S FUNCTIONS

• A Green’s function is the response to a unit impulse at a location and time, and superposition expresses a source-driven response through the kernel.The displayed formulation assumes zero initial data and homogeneous Neumann boundary conditions; the full setting also treats initial and boundary inputs.Source: §3 · p. 4
• For a fixed medium, linearity lets one Green’s kernel predict responses to different source histories, but the kernel changes when the medium changes.The kernel is available analytically only in a few simple cases.Source: §3 · p. 4
• The paper studies parabolic systems with transport field b and spatially varying diffusivity κ; b = 0 gives heat conduction and nonzero b gives advection–diffusion.Under the stated standard assumptions, the problem admits a Green’s kernel continuous up to the boundary for positive time.Source: §3 · p. 4

4.1 RESTRICTING THE AMBIENT GREEN KERNEL TO SUBMANIFOLDS

• GObO learns only the ambient Green kernel’s restriction from the source manifold to the observation manifold, rather than modeling arbitrary point pairs in the full domain.The source integral is expressed in chart coordinates; the source surface element is absorbed into the restricted kernel.Source: §4.1 · pp. 4–5

4.2 SPECTRAL SEPARATION OF TIME AND SPACE IN THE RESTRICTED KERNEL

• A naive pointwise kernel parametrization would require more than 10^9 evaluations for one example prediction, motivating spectral separation of time and space.The example uses a 2D domain with 64 points per spatial axis and 64 time steps.Source: §4.2 · p. 5

4.3 EXPONENTIAL CONVERGENCE OF SPECTRAL TRUNCATION FOR THE HEAT KERNEL

• For the heat equation, truncation error for the restricted kernel decays exponentially in M^(2/d) once time stays at least τ0 > 0 from the singularity.The bound is stated for each fixed medium and positive τ0.Source: §4.3 · pp. 5–6

4.4 FINITE-DIMENSIONAL PARAMETRIZATION OF THE RESTRICTED KERNEL

• Finite-dimensional bases represent the source- and observation-side spatial factors, while a conditioning network maps the medium to their mode coefficients.Higher-rank factorization is used as an optimization heuristic because rank-one factors may stall under assumptions not fully satisfied by this problem.Source: §4.4 · p. 6

4.5 MARKOVIAN STREAMING VIA AN AUGMENTED KERNEL STATE

• For piecewise-constant sources on a uniform time grid, the augmented kernel state gives exact streaming evaluation with fixed-dimensional memory independent of prediction horizon.The recurrence does not amplify perturbations when the rates have nonnegative real parts; strictly positive real parts yield uniformly bounded states for bounded projected sources.Source: §4.5 · p. 6

4.6 REUSING THE LINEAR KERNEL FOR MILD NONLINEARITIES

• Source-like nonlinearities can be handled by alternating linear propagation with nonlinear source updates, with streaming making repeated updates inexpensive.The paper develops this construction for radiative heat loss.Source: §4.6 · p. 7
• For nonlinear heat conduction, the Kirchhoff transform can reduce the problem exactly to the learned linear response or yield a transformed nonlinear system advanced approximately with the streaming state.Source: §4.6 · p. 7

5.1 EXPERIMENTAL SETTING

• The experiments cover three-dimensional heat conduction and advection–diffusion, evaluating mean relative error and standard deviation across three seeds and unseen inputs and media.The models are trained on static source pulses and tested on held-out static pulses and sources moving along a straight line.Source: §5 · p. 7, §5.1 · p. 7

5.2 GOBO LEARNS A KERNEL THAT GENERALIZES ACROSS SOURCES

• FNO-2D's error rises about 100× on moving sources while GObO changes only slightly; adapted NGO transfers stably but has errors around 4× higher than GObO.The same pattern holds across additional configurations: FNO-2D error rises about 4–7× while GObO remains stable.Source: Table 2 · p. 7, §5.2 · p. 7
• Figure 2 shows GObO tracking the moving response closely, while medium-conditioned FNO-2D misses its location, shape, and magnitude.The figure compares the true final temperature and source trajectory, temperature profiles at three times, and peak position and magnitude.Source: §5.2 · p. 7, Figure 2 · p. 8

5.3 THE AUGMENTED KERNEL STATE CAPTURES NON-CLOSED SUBMANIFOLD DYNAMICS

• About 3× more observation history reduces restricted-trace next-step error; transient NGO has about 3.5× higher rollout error than GObO, and a purely spatial GObO also benefits from history.A closed two-dimensional heat system is predicted to machine precision from a single state; full GObO carries the missing information in an augmented kernel state.Source: Figure 3 · p. 8, §5.3 · p. 8

5.4 THE KERNEL PARAMETRIZATION CAPTURES THE SPATIOTEMPORAL STRUCTURE

• A naive space-time tensor decomposition raises error by 3–4×; in a controlled kernel-fitting example, increasing rank L from 1 to 8 reduces the objective by about 14×.The rank sweep shows little improvement beyond 8, motivating L = 16 as a safety margin.Source: §5.4 · pp. 8–9

5.5 GOBO ENABLES RAPID SOURCE EVALUATION AND NONLINEAR CORRECTIONS

• After one per-medium kernel construction, source evaluations are 134× faster on CPU and 5.6× faster on GPU; at 2× training resolution, additional error is numerically negligible.Without retraining, nonlinear corrections improve accuracy by more than 11× for radiative feedback and up to 29× for temperature-dependent conductivity.Source: Table 3 · p. 9, §5.5 · p. 9

• Neural operators can map sources on one submanifold to observations on another, but generally treat the relation as a black box and do not explicitly impose source linearity.GObO encodes that structure with a reusable restricted Green kernel.Source: §6 · p. 9
• Methods for PDEs posed intrinsically on manifolds differ from GObO's setting, where ambient-domain dynamics couple the source and observation submanifolds.The observed submanifold is not closed in this setting.Source: §6 · p. 9
• Unlike learned Green-function approaches based on steady-state representations or fixed time-stepping schemes, GObO learns a continuous time-dependent kernel between submanifolds.The passage also contrasts with methods that identify a fixed operator or condition representations on geometry or PDE coefficients.Source: §6 · p. 9

7 CONCLUSION

• GObO maps directly between source and observation submanifolds, preserving source linearity and causality with horizon-independent streaming, and trades some in-distribution accuracy for reliable source transfer and cheap reuse.The conclusion also reports resolution transfer and extensions to mild nonlinearities without retraining the linear kernel.Source: §7 · p. 9

OVERVIEW OF THE APPENDIX

• The appendices cover related work, proofs and derivations, equations and numerical references, architecture and protocols, experiment procedures, and complete results and ablations.Source: OVERVIEW OF THE APPENDIX · p. 14

A.1 OPERATOR LEARNING FOR PDES

• The operator-learning discussion situates GObO within function-space PDE models, including DeepONet and FNO, while taking measured submanifolds as outputs.Source: §A.1 · p. 14

A.2 PARTIAL OBSERVATIONS AND REDUCED REPRESENTATIONS

• The partial-observation discussion contrasts compact latent or reduced-state approaches with GObO's use of the observed submanifold as prediction space and the ambient medium to determine response.The nonlinear construction is also connected to projection-based model reduction.Source: §A.2 · p. 15

A.3 GEOMETRY AND PHYSICAL CONTEXT

• The geometry discussion contrasts GINO's full-field supervision and full-domain latent states with GObO's manifold observations and explicit response kernel.Source: §A.3 · p. 15

A.4 NONLINEAR METHODS BUILT ON LINEAR SOLVERS

• Streaming representations let exponential integrators advance the linear state exactly, while source linearity evaluates nonlinear-method intermediate forcings using the learned kernel.Operator splitting, implicit–explicit methods, exponential integrators, and the Kirchhoff transform all rely on access to a linear solver.Source: §A.4 · p. 15

A.5 GREEN’S-FUNCTION AND KERNEL-BASED OPERATORS

• Pointwise kernel methods require repeated coordinate queries, whereas structured methods produce a reusable finite representation applied after one inference or identification step.The appendix notes that dense space–time kernel recovery by pointwise queries is impractical for the three-dimensional transient setting considered.Source: §A.5 · p. 15, Table 4 · p. 15
• GObO conditions each new medium once to produce a structured temporal-lag kernel, enabling direct source convolution and finite streaming without pointwise network queries or ambient-state reconstruction.Unlike the closest baseline, NGO, it does not use a one-step treatment of time; partial observations are not assumed to form a closed state.Source: §A.5 · p. 16

B.1 GREEN REPRESENTATIONS FOR INITIAL, INTERIOR, AND BOUNDARY DATA

• For manifold-restricted sources, the Green-kernel integral has a uniform short-time bound proportional to τ^-1/2, while separated source and observation manifolds eliminate that singularity.Initial and interior forcing share a kernel for time-independent systems; boundary inputs may use a different kernel depending on the boundary operator, and their responses add.Source: §B.1 · pp. 17–18

B.2 APPROXIMATION OF THE RESTRICTED KERNEL

• The restricted heat-kernel eigenfunction expansion converges uniformly for positive time lags and admits a truncation bound, yielding the stated L2 approximation bound.The theorem assumes zero advection, compact parameter sets, and bi-Lipschitz C1 source and observation charts.Source: §B.2 · pp. 19, 21

B.3 OPTIMIZATION WITH ADDITIONAL SEARCH RANK

• The additional-search-rank optimization analysis is only qualitative: the temporal design matrix has condition number 5.8 × 10^14, and finite-run loss cannot distinguish plateaus from poor local minima.The cited overparameterization results do not guarantee global optimality for this rectangular, multi-rate factorization, and the required assumptions have not been established for its ill-conditioned temporal design.Source: §B.3 · pp. 21–22
• At L = 1, attained loss remains more than an order of magnitude above the convex coefficient-space optimum, while L ≥ 8 brings it within approximately 1%.Increasing L changes both the representable coefficient matrices and the optimization landscape; a theoretical characterization of the observed rank dependence is left to future work.Source: §B.3 · p. 22

B.4 EXACT STREAMING EVALUATION

• For piecewise-constant sources, the streaming recurrence and readout exactly match batch evaluation of the kernel.The recurrence is defined on a uniform time grid using projected source values.Source: §B.4 · pp. 22–23
• When Re(λk) ≥ 0, perturbations between states driven by the same source remain uniformly bounded.The stability condition follows from |e^−λk∆| ≤ 1.Source: §B.4 · pp. 24–25

B.5 EXTENSIONS TO MILD NONLINEARITIES

• The nonlinear extensions reuse the learned linear response, either by alternating propagation with a feedback source or by transforming the heat equation.The reference medium remains fixed in both constructions.Source: §B.5 · p. 25

B.5.1 RADIATIVE BOUNDARY FEEDBACK

• Radiative loss is treated as a temperature-dependent feedback source, alternated with linear propagation using a semi-implicit update.The streaming representation replaces repeated full-history convolution with a fixed-cost state recurrence and a small coefficient-space solve.Source: §B.5.1 · pp. 25–26

B.5.2 KIRCHHOFF TRANSFORMATION

• The Kirchhoff transform exactly linearizes the case where conductivity and heat capacity vary by the same factor, but the conductivity-only case retains a multiplier.Because that multiplier is volumetric while predictions are traces, the latter case uses a trace-based approximation advanced with the streaming state.Source: §B.5.2 · pp. 26–27

C DATASETS

• The dataset section defines the equations, media, sources, observation manifolds, and numerical reference solutions used in the experiments.Experiment-specific model definitions and protocols follow in later sections.Source: C DATASETS · p. 29

C.1 HEAT-CONDUCTION DATA

• The heat data use a cube with homogeneous Neumann conditions, a top-face surface flux represented in the top cell layer, and temperature observed in that layer.Linear and radiative reference data use different Crank–Nicolson substep counts.Source: §C.1 · p. 29
• Collocated heat experiments use a two-phase medium, whereas interior-plane experiments use a graded medium.The two-phase medium assigns distinct solid and powder conductivities; the graded medium spans 10^-4 to 10^-2.Source: Table 6 · p. 29, §C.1 · p. 29
• Static Gaussian heat pulses train the models, while straight and serpentine moving scans form test sets.Training pulse centers are sampled within surface margins, with durations of 6–64 frames; only static pulses are used for training.Source: §C.1 · pp. 29–30

C.2 DISTINCT OBSERVATION SURFACES

• Horizontal and vertical interior-plane datasets use identical media and sources to isolate observation geometry, with 1024 training samples per plane and 128 per test class.Each plane has static, straight-scan, and serpentine test classes.Source: §C.2 · p. 30

C.3 ADVECTION–DIFFUSION DATA

• The advection–diffusion setup places the source on the ground plane and observes concentration on a sensor curtain at the domain wall.The domain is [0, 9 km]^2 × [0, 2 km], and the reference uses 32 substeps per frame.Source: §C.3 · pp. 30–31
• Static fumes train the models, while moving tests use one crosswind track or two to four alternating crosswind legs.Moving emitters travel at 1.2–2.4 m/s with constant amplitude and radius; static fumes must exceed a curtain RMS threshold.Source: §C.3 · p. 31

C.4 NONLINEAR AND RESOLUTION-TRANSFER DATA

• Nonlinear experiments include radiative boundary loss and temperature-dependent conductivity, with static and shifted-source test sets.Shifted sources include scans, rasters, or sums of two static pulses.Source: §C.4 · p. 31
• The resolution study pairs 64 samples across 64^3 and 128^3 grids, matching the fine target to the same physical slab as the coarse target.The medium is prolonged as a piecewise-constant field and the same physical source is evaluated on each grid.Source: §C.4 · p. 31

D.1 COSINE BASIS AND KERNEL REPRESENTATION

• A cell-centered cosine basis represents source and observation charts, and its continuous chart coordinates permit coefficients to be evaluated on another grid.The retained functions form a 32 × 32 coefficient grid with 1024 source and observation coefficients.Source: §D.1 · pp. 31–32

D.2 MEDIUM-CONDITIONING NETWORK

• The conditioning network maps the medium to kernel factors before the source is applied, so later sources in that medium require no further network evaluation.GObO produces low-rank factors rather than the dense coefficient matrix used by the adapted NGO system network.Source: §D.2 · pp. 32–33, Figure 4 · p. 33

D.3 TEMPORAL PROPAGATION AND STREAMING STATE

• At inference, rate-wise source states are propagated and synthesized into observations; the heat model uses 256 scalar state coefficients and agrees with dense evaluation to floating-point precision.Changing frame spacing changes only the analytical propagation weights.Source: §D.3 · pp. 33–34

D.4 ADVECTION–DIFFUSION VARIANT

• The advection–diffusion variant uses complex rates and conjugate terms so the synthesized curtain field is real-valued.The rate real part represents diffusive decay, while its imaginary part represents advective phase transport.Source: §D.4 · p. 34

D.5 TRAINING AND EVALUATION

• The reported evaluations use shared data splits, with normalization and augmentation choices adapted to each geometry and model.GObO receives sources in physical units because it applies them after medium conditioning.Source: §D.5 · p. 34

E EXPERIMENTAL DETAILS

• Models share data splits, loss, and evaluation metrics; heat models train on static pulses, while advection–diffusion models train on stationary fumes and test moving sources without retraining.The comparisons include direct-output baselines, adapted NGO, and one-step variants.Source: §E.1.1 · p. 35, §E.2.1 · p. 35, §E.2.2 · p. 35

F.2.1 HEAT SOURCE TRANSFER

• Direct models perform best on static heat sources but degrade sharply under motion, while GObO changes little across the three source classes.All heat models are trained on static pulses and evaluated without retraining on held-out pulses and moving scans.Source: §E.2.1 · p. 35, §F.2.1 · p. 39, Table 17 · p. 39

F.2.2 BASELINE CAPACITY AND GOBO DESIGN

• Deeper direct-output baselines do not fix moving-source transfer; fixed rates, shared factors, and search rank L = 16 give the most reliable GObO results.The controls support the analytic fixed-rate design and show that search rank 1 is insufficient.Source: §F.2.2 · p. 39, Table 18 · p. 39

F.2.3 DISTINCT OBSERVATION SURFACES

• GObO’s source-transfer advantage persists when source and observation surfaces are separated, including an observation plane orthogonal to the source face.The vertical-plane aggregate gap persists throughout the moving scan.Source: §F.2.3 · p. 40, Table 20 · p. 40, Figure 6 · p. 41

F.2.4 ADVECTION–DIFFUSION SOURCE TRANSFER

• Fixed physical rates keep GObO close to its static error under both advection–diffusion source shifts; learned rates degrade, and FNO-2D does not transfer to motion.The curtain response supports the quantitative comparison without showing the catastrophic qualitative failure reported for heat.Source: §F.2.4 · p. 41, Table 21 · p. 41, Figure 7 · p. 41

F.3.1 ADAPTED TRANSIENT NGO

• The adapted NGO with S = 1 reaches 1-step error 0.012 and rollout errors 0.039, 0.043, and 0.027 on static, straight, and serpentine sources.The published S = 0.8 threshold clips every sample and gives static rollout error 0.721.Source: §F.3.1 · p. 42

F.3.2 ONE-STEP PREDICTION FROM PARTIAL OBSERVATIONS

• The fully observed control has residual 4 × 10−15, versus 3.2 × 10−3 at k = 1 and 1.1 × 10−3 at k = 16 on the three-dimensional trace.The contrast supports that partial observations do not form a closed state; longer learned histories can help within the training window but become unstable beyond it.Source: §F.3.2 · p. 42, Table 23 · p. 42

F.4.2 NAIVE TENSOR-DECOMPOSITION ACCURACY

• Removing the lag law raises static error from 0.011 to 0.035 and moving error from 0.012 to 0.045, while producing substantial pre-source response.The impulse-response comparison shows the lag-dependent kernel captures decay and onset that the generic space–time basis misses.Source: §F.4.2 · p. 43, §F.4.1 · p. 43, Table 24 · p. 43

F.4.3 ARCHITECTURE AND MEDIUM-PATHWAY ABLATIONS

• Replacing the learned volume encoder with fixed medium projections raises static error to 0.103 at similar parameter count, showing that basis size alone does not explain the pathway’s contribution.An MLP on the same projections reaches 0.020 but uses 137.23 million parameters.Source: §F.4.3 · p. 43, Table 25 · p. 44

F.4.4 CONTROLLED CURVED-SURFACE RECOVERY

• On distinct curved surfaces, the recovered kernel has slice errors 9.4 × 10−3, 4.3 × 10−3, and 1.6 × 10−3 at lags 0.05, 0.075, and 0.125.Held-out error on 64 Gaussian chart sources is 0.027.Source: §F.4.4 · p. 44, Figure 9 · p. 44

F.4.5 FACTOR-RANK SWEEP

• For factor search rank L ≥ 8, the controlled fit is within approximately one percent of the coefficient-space optimum; L = 16 is used in the heat experiments.At L = 1, the objective remains more than an order of magnitude above the optimum; the recovered rank remains 7 from L = 8 through L = 64.Source: §F.4.5 · p. 44, Table 26 · p. 45

F.5.1 REPEATED-SOURCE TIMING

• Dense and streaming evaluation agree to 5.2 × 10−7, and the matched-accuracy finite-volume setting has relative error 0.0096 against the production solve.Conditioning is paid once; subsequent source queries use the explicit kernel without another network pass.Source: §F.5.1 · pp. 44–45, Table 27 · p. 46, Table 28 · p. 46

F.5.2 RESOLUTION TRANSFER

• Coarse conditioning followed by a fine-grid query and coarse computation followed by fine synthesis both reach the 0.026 reference discretization gap.The coarse-conditioning reuse path is specific to GObO; operator-reuse paths avoid fine-grid encoder evaluation and are unaffected by positional-encoding choice.Source: §F.5.2 · p. 45, Figure 10 · p. 45, Table 29 · p. 47

F.5.3 RADIATIVE FEEDBACK

• Both radiative corrections reach relative error 0.027 ± 0.001 over 128 samples and three seeds, and remove the uncorrected kernel’s peak overshoot without retraining.The explicit and linearized implicit updates reuse the unchanged GObO kernel.Source: §F.5.3 · pp. 45–46, Figure 11 · p. 46

F.5.4 TEMPERATURE-DEPENDENT CONDUCTION

• The exact Kirchhoff transform keeps GObO near its linear accuracy across conductivity contrasts and shifted sources; the trace-based update improves every tested setting without retraining.The corrections preserve the source-transfer benefit while accounting for the nonlinear material response.Source: §F.5.4 · p. 46, Table 31 · p. 48, Figure 12 · p. 48

F.5.5 FIXED-POINT COMPARISON

• GObO improves substantially under both fixed-point residual constructions, while direct FNO remains inaccurate on shifted sources; the remaining shifted-source error is attributed to the surrogate, not outer-solve failure.Table 32 separates outer-iteration convergence from surrogate accuracy on the effective sources generated by the iteration.Source: §F.5.5 · p. 47, Table 32 · p. 48