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Bi-FORK: Generative Modeling of High-Dimensional Bifurcating Systems

Anna Zimmel, Fleur Hendriks, Markus Holzleitner, Florian Sestak, Martin Weichselbaumer, Vlado Menkovski, Johannes Brandstetter

arXiv:2610.12449v1cs.LGcs.AIcs.CEphysics.comp-ph

TL;DR

Symmetry-breaking bifurcations map a single condition to multiple equally valid solutions, a setting conventional surrogates and evaluation metrics do not adequately address. Bi-FORK generates coherent full trajectories with latent flow matching and repulsion-guided sampling. Across three physical systems, it recovers multimodal solutions at scales up to 260,000 points, several orders of magnitude beyond prior work.

  • Problem

    Symmetry-breaking bifurcations create multiple equally valid solutions for one input, while standard prediction and evaluation approaches do not capture their multimodal, trajectory-level structure.

    Source: §1 · pp. 1–2

  • Method

    Bi-FORK uses latent flow matching to generate full trajectories jointly and particle guidance to repel concurrent samples toward distinct solutions.

    Source: §1 · p. 2, §2.3 · p. 5

  • Results

    Across buckling beams, mechanical metamaterials, and Allen-Cahn phase separation, Bi-FORK recovers multimodal solutions at up to 260,000 points, several orders of magnitude beyond prior work.

    Source: §1 · p. 2, §4 · p. 10

  • Takeaways & Limitations

    Bi-FORK extends generative modeling of bifurcating physical systems to large discretizations while recovering their multimodal solution structure.

    Source: §4 · p. 10

    Particle guidance can trade accuracy for diversity by pushing samples off the solution manifold, and Allen-Cahn coverage estimates rely on four simulated realizations per condition and sign flips.

    Source: §4 · p. 9

Abstract

from arXiv · show

Bifurcations are ubiquitous in physical systems, from structural buckling to fluid and climate dynamics, yet they remain largely unexplored in deep learning. At a symmetry-breaking bifurcation, a single input admits multiple equally valid solutions, violating the one-to-one assumption underlying most learned physical surrogates. We introduce Bi-FORK, a generative framework for learning these one-to-many solution maps in high-dimensional systems. Bi-FORK generates complete trajectories through latent flow matching, preserving space and time coherence, and uses repulsion-guided sampling to recover distinct solution branches in a single amortized pass. We evaluate Bi-FORK on buckling beams, mechanical metamaterials, and Allen-Cahn phase separation, spanning continuous, discrete, and field-valued bifurcations with discretizations up to 260,000 points. Bi-FORK recovers the multimodal solution structure while scaling several orders of magnitude beyond prior approaches, opening generative modeling to high-dimensional bifurcating physical systems.

1 Introduction

Symmetry-breaking bifurcations create one-to-many solution maps that challenge conventional learned surrogates and make coherent distributional evaluation difficult. Bi-FORK addresses these challenges with whole-trajectory latent flow matching and repulsion-guided sampling, and is evaluated across three systems at scales up to 260,000 points.

1 Introduction

  • Symmetry-breaking bifurcations make one input correspond to multiple equally valid solutions, while discriminative models average over them and produce nonphysical predictions.Classical methods can compute solutions accurately, but recovering high-dimensional bifurcation structure requires repeated continuation or deflation solves alongside stability and mode analyses.Source: §1 · p. 1
  • Existing direct approaches to bifurcation structure handle at most a few hundred degrees of freedom and small, known, discrete symmetry groups [17] [61] [30] [25].Most probabilistic physical surrogates instead model epistemic or unresolved uncertainty [19, 60, 12].Source: §1 · p. 2
  • Global branch coordination is difficult for frame-wise autoregressive models and local-message-passing GNNs, especially when bifurcations are rare in trajectories [43] [13] [11] [5].A trajectory that switches branches midway or realizes inconsistent branches across its elements is invalid.Source: §1 · p. 2
  • Evaluating bifurcating predictions is difficult because pointwise metrics miss trajectory coherence, best-of-K ignores most of the distribution, and generic two-sample distances are hard to estimate at high dimensions.Generic distances also do not exploit the symmetry structure of bifurcating solutions [9].Source: §1 · p. 2
  • Bi-FORK jointly generates full trajectories in latent space and uses particle guidance to recover distinct solutions in one pass, with symmetry-aware diagnostics for distribution coverage.Whole-trajectory generation keeps branch choices consistent across space and time; particle guidance acts as a parallelized analog of classical deflation.Source: §1 · pp. 2–3
  • The evaluation spans buckling beams, mechanical metamaterials, and Allen-Cahn phase separation, reaching discretizations up to 260,000 points and outperforming existing approaches in accuracy and efficiency.The evaluated systems include grid- and mesh-based problems, discrete and continuous symmetries, and trajectories of 32 to 200 timesteps.Source: §1 · p. 2

2 A Blueprint for Bifurcating Regimes

The section formalizes symmetry-breaking bifurcations as conditions with multiple admissible trajectories and describes how to model their distributions. It presents latent trajectory modeling, repulsion-guided sampling, and metrics for trajectory precision and mode-frequency fidelity.

2.1 Bifurcation Systems

  • Bifurcations mark loss of equilibrium uniqueness, with discrete, continuous, and field-valued mode sets across the considered systems.Symmetry organizes outcomes into orbits; Allen-Cahn mode sets can comprise a union of orbits.Source: §2.1 · pp. 3–4

2.2 Learning task

  • For each condition, the learning task is to predict a distribution over feasible trajectories supported on its mode set.For single-orbit datasets, the target is uniform over modes under the assumption that unknown perturbations have no preferred direction.Source: §2.2 · p. 4

2.3 Modeling

  • Point-cloud states are encoded into fixed learned latent tokens and decoded at the same reference positions, anchoring spatial reconstruction to the input geometry.The latent trajectory can then be analyzed and predicted independently of pointwise spatial reconstruction.Source: §2.3 · p. 5
  • Latent flow matching maps noise to complete trajectories, while joint particle sampling adds repulsion at predicted post-bifurcation frames to spread samples across branches.The repulsive term acts on masked frames; samples that would otherwise collapse onto one mode are pushed apart.Source: §2.3 · pp. 5–6

2.4 Metrics

  • MCon-MAE measures trajectory error after assigning samples to their nearest ground-truth mode, while JSD measures predicted-versus-ground-truth mode-frequency fidelity.The metrics complement one another by assessing trajectory-level precision and relative outcome frequencies.Source: §2.4 · p. 6

3 Experiments

The experiments assess Bi-FORK on continuous beam buckling, discrete metamaterial modes, and Allen–Cahn fields. Results report branch recovery alongside trajectory validity, PDE consistency, and inference cost.

3.1 Beam3D

  • Beam3D inference uses particle guidance to spread samples over the admissible post-buckling manifold, and reports rejection rate alongside MCon-MAE and JSD.The rejection rate is the fraction of trials whose mean displacement from their matched mode exceeds a threshold; MCon-MAE and JSD include all trials.Source: §3.1 · p. 7
  • Bi-FORK substantially improves MCon-MAE with 0% trajectory rejection and a highly competitive JSD, closing the gap with GeoTDM.It generates 180 Beam3D samples for one condition in 3.95 seconds, approximately 18× faster than STFlow and 1,966× faster than GeoTDM.Source: Figure 5 · p. 6, §3.1 · p. 7

3.2 Mechanical Metamaterials

  • For metamaterials, sampling omits the repulsive term when |Mc| = 1; JSD and coverage are reported only for |Mc| ∈{2, 4}.The method samples 16 times per condition to estimate the distribution.Source: §3.2 · p. 8
  • Bi-FORK improves accuracy and distributional coverage over the affine baseline, retrieving all four modes in roughly 50% of sampled test cases.For the illustrated two- and four-mode geometries, Bi-FORK produces a nearly uniform mode distribution.Source: Figure 6 · p. 7, §3.2 · p. 8, Table 2 · p. 8

3.3 Allen-Cahn

  • Allen–Cahn uses a ViT to encode the periodic field and generates the full rollout from ϵ and µ, applying particle guidance at every valid frame.Its evaluation separates mode-coverage, translation, and rotoreflection JSD; MCon-MAE measures distance to a PDE-relaxed solution.Source: §3.3 · p. 9
  • 0.045 MCon-MAE (versus 0.052 without particle guidance) and 0.089 mode JSD (versus 0.131) are reported for Bi-FORK.Translation and rotoreflection JSDs are identical for both variants.Source: §3.3 · p. 9

4 Discussion

Bi-FORK recovers multimodal solutions in large bifurcating systems, while its guidance, evaluation, and symmetry handling retain important limitations. The Allen–Cahn coverage metric does not establish coverage of the joint distribution.

Examples

  • Figure 8 compares ground-truth and predicted samples for Allen–Cahn at µ = 0.86 and ϵ = 0.094.Source: Figure 8 · p. 9

Limitations

  • Particle guidance can trade accuracy for diversity because repulsion may push samples in the same valid mode off the solution manifold.The Allen–Cahn mode-coverage JSD uses four simulated realizations per condition and their sign flips; separate JSDs do not guarantee joint-distribution coverage.Source: §4 · p. 9
  • The model learns symmetry from data and augmentation rather than enforcing it architecturally.Source: §4 · p. 10

Conclusion

  • Bi-FORK recovers multimodal solutions across three system types at discretizations up to 260,000 points, several orders of magnitude beyond prior work.The largest systems are compared with a deterministic affine baseline and the authors’ ablations because no existing stochastic method scales to that size.Source: §4 · p. 10

AI use statement

The paper discloses generative-AI assistance for implementation, data preparation, literature identification, and manuscript and figure polishing, while assigning verification and responsibility to the authors. It also documents reproducibility resources, notation, and additional bifurcation context.

AI use statement

  • AI tools assisted with baseline and evaluation-code implementation, dataset cleaning and reformatting, and ideas for mathematical claims; authors reviewed or verified these outputs.The proposed method was implemented by the authors, and the tools were not used to generate synthetic data, design experiments, or interpret results.Source: AI use statement · p. 10
  • AI tools also helped identify literature, polish readability, and make cosmetic figure edits, with authors checking papers, technical edits, and underlying figure numbers.Source: AI use statement · p. 10

Reproducibility statement

  • The paper points to appendices for dataset splits, model and training details, and dataset-specific evaluation pipelines, and provides a code repository.Source: Reproducibility statement · p. 10

A Notation

  • Table 3 summarizes notation, with dataset-specific quantities defined where they are used.The notation spans conditions and modes, latent representations and flow matching, sampling and guidance, losses, and evaluation metrics.Source: A Notation · p. 15, Table 3 · p. 15
  • The learned trajectory distribution is conditioned on the problem condition, while the latent flow maps noisy latent trajectories toward predicted clean trajectories.Source: A Notation · p. 15

B Additional bifurcation context

  • Bifurcation time is estimated from pairwise trajectory divergence, taking the first timestep when divergence exceeds a small threshold; for Allen–Cahn the estimate is per orbit.For quasi-static datasets it recovers bifurcation time up to threshold resolution; in Allen–Cahn it marks macroscopic domain formation.Source: B Additional bifurcation context · p. 16
  • The paper distinguishes structured branching into symmetry-related outcomes from chaotic divergence, framing the learned target as a multimodal distribution over valid branches.Source: B Additional bifurcation context · p. 16
  • The mechanical setting concerns equilibrium branch splitting at loss of uniqueness and focuses on generic supercritical branching; sudden subcritical or imperfect-buckling jumps are out of scope.Allen–Cahn is treated as genuinely time-dependent, using the dynamical version of bifurcation theory.Source: B Additional bifurcation context · p. 16

C Datasets

The datasets cover continuous-direction beam buckling and discrete post-buckling modes in mechanical metamaterials. Their setups provide distinct bifurcation structures, from infinitely many beam directions to hierarchical two- and four-mode branching.

C.1 Beam3D

  • Beam3D applies incremental downward tip displacement to beams with 3 to 11 nodes; beyond a critical displacement they buckle in any direction ϕ in the x-y plane, yielding infinitely many modes under identical loading.The dataset contains 997 trajectories of up to 200 timesteps.Source: §C.1 · pp. 16–17
  • Beam3D is suited to studying continuous symmetry-related modes, and Figure 9 shows trajectories with different buckling directions across node counts.Source: §C.1 · p. 17, Figure 9 · p. 17

C.2 Mechanical Metamaterials

  • The mechanical metamaterials dataset contains 12,240 trajectories from 1,020 geometries across all 17 wallpaper groups, with meshes of 1,000–36,000 points per timestep.Each geometry is simulated under 12 loading paths, with trajectories up to 32 timesteps.Source: §C.2 · p. 18
  • Metamaterial geometries can end in one, two, or four post-buckling modes; examples show mirror-reflected branches and a four-mode outcome produced by two successive bifurcations.The data split assigns each geometry, along with all its loading paths, to one subset to prevent geometry leakage.Source: §C.2 · p. 18

C.3 Allen-Cahn Dataset

The Allen–Cahn dataset models phase separation in a three-dimensional periodic domain, with parameters controlling interface properties and local phase structure. Four simulations and sign-flipped counterparts provide candidate modes for each condition.

C.3 Allen-Cahn Dataset

  • The Allen–Cahn state is a scalar phase field on a three-dimensional periodic domain; ϵ controls interface width and smoothness, while µ controls local phase structure.For µ > 0, perturbations of the mixed state grow toward domains near u = ±√µ; the sampled µ range also includes a small non-separating regime with µ < 0.Source: §C.3 · p. 19
  • The dataset contains 1,000 parameter conditions simulated on a 64^3 grid, with 51 snapshots over [0, 50] from trajectories integrated for 5,000 steps.Every trajectory starts from the homogeneous state and receives random noise of amplitude 0.1; the data use periodic boundary conditions.Source: §C.3 · p. 19
  • Each (ϵ, µ) condition has four simulated phase-separation trajectories, and sign inversion supplies four phase-exchanged counterparts, yielding up to eight candidate modes.Conditions, rather than individual trajectories, are split among training, validation, and test sets to prevent leakage.Source: §C.3 · p. 19, Table 7 · p. 20

D.1 Architecture

The architecture compresses physical states into latent representations, then models complete latent trajectories with a flow model before decoding them back to physical signals.

D.1 Architecture

  • A Perceiver-style cross-attention autoencoder encodes nodal states into fixed learned tokens and decodes positional queries into per-point deformations for mechanical metamaterials and Beam3D.The latent size is independent of the number of mesh nodes.Source: §D.1 · p. 20
  • Allen-Cahn uses a ViT3D that patchifies the voxel grid into 8 × 8 × 8 patches and encodes them into a 4 × 4 × 4 latent grid, which its decoder upsamples.Source: §D.1 · p. 20
  • Dataset-statistic normalization of physical fields and conditioning parameters, alongside layer normalization, stabilizes latents without variational regularization.Source: §D.1 · p. 20
  • The two-stage model encodes physical states and reconstructs them with a decoder, then freezes the autoencoder while a SiT learns the latent-trajectory distribution from Gaussian-coupled trajectories.The predicted latent trajectory is decoded back to the physical signal.Source: §D.1 · p. 20
  • Beam3D and metamaterials use a generalized variance-preserving path, while Allen-Cahn uses a linear path; the configured models estimate the clean latent trajectory directly.Source: §D.1 · p. 20

D.2 Implementation Details

Implementation choices tailor the latent representation, conditioning, and symmetry handling to each physical system.

D.2 Implementation Details

  • Beam3D states encode three-dimensional displacement fields in six 128-dimensional Perceiver tokens, independent of mesh-node count, conditioned on tip displacement and the initial frame.Random azimuthal rotations expose the model to the beam’s SO(2) orbit of equivalent buckling directions.Source: §D.2 · p. 21
  • Mechanical metamaterial states combine displacement, node positions and types, and wallpaper-group information, then compress to 128 tokens of dimension 128 conditioned on the deformation gradient.Optimal-transport matching is restricted to trajectories under the same physical condition to avoid coupling different conditions.Source: §D.2 · p. 21
  • Allen-Cahn fields use a ViT3D on a periodic 64^3 grid, conditioned on (ϵ, µ), with temporal and spatial rotary embeddings and symmetry-preserving translation, sign, reflection, and permutation transformations.Source: §D.2 · p. 21

D.3 Training

Training separates physical-state reconstruction from latent trajectory modeling, with masked losses and particle-guidance adaptations tailored to bifurcating trajectories.

D.3 Training

  • Training first optimizes an autoencoder, then freezes it while fitting the latent flow model and conditioner using Gaussian-noise coupling, optionally via condition-restricted optimal transport.Invalid spatial elements and timesteps are excluded; standard models observe the first frame, whereas Allen-Cahn transfer generation is unconditional.Source: §D.3 · p. 21
  • Beam3D trajectories receive random longitudinal-axis rotations and Allen-Cahn trajectories symmetry transformations applied consistently across frames; metamaterials use no generic spatial augmentation.Source: §D.3 · p. 21

D.4 Loss Function

  • The autoencoder uses node-masked reconstruction MSE, while latent-flow training uses frame-masked trajectory MSE and may add temporal pre/post-bifurcation classification for Beam3D and metamaterials.Standard Allen-Cahn omits the auxiliary classification head and uses only the flow loss.Source: §D.4 · p. 22

D.5 Adaptations to particle guidance

  • Particle guidance computes repulsion on denoised latent estimates, scales it by denoising confidence, and uses a normalized kernel-weighted average of unit pairwise directions.Normalization by total kernel mass keeps force scale stable as neighboring particles change.Source: §D.5 · p. 22
  • Guidance excludes self-interaction from the force numerator but retains it in the denominator, and applies repulsion only at predicted bifurcation frames with a bounded schedule that strengthens toward the data endpoint.The schedule never switches off.Source: §D.5 · p. 23

D.6 Hyperparameters

D.7 Evaluation Details: Beam3D and Metamaterials

Evaluation matches generated trajectories to symmetry-aware modes and assesses both physical consistency and distributional coverage. The reported metamaterial comparison finds JSD below the finite-sample uniform reference, while the tables summarize multiple datasets and metrics.

D.7 Evaluation Details: Beam3D and Metamaterials

  • For discrete-mode metamaterials, rollouts are matched to nearest ground-truth modes using a whole-trajectory distance threshold; Beam3D continuous modes are evaluated by buckling azimuth and orbit matching.The Beam3D setup rejects trajectories beyond a relative-distance threshold and uses 180 trials with 10 integration steps for comparisons.Source: §D.7 · pp. 23–24

D.8 Evaluation Details: Allen-Cahn

  • Allen-Cahn evaluation first matches generated fields to branches using a translation- and point-group-invariant radial power spectrum, then exhaustively aligns orientation and periodic shift.The alignment searches the 48-element octahedral group; the exact minimum is found directly.Source: §D.8 · p. 24
  • Allen-Cahn distributional evaluation separates mode coverage, translation, and rotoreflection JSD components, while its MCon-MAE compares each raw rollout with its own relaxed PDE-consistent projection.The MCon-MAE calculation uses no saved reference rollout, matched branch, or branch label.Source: §D.8 · p. 24

E.1 Quantitative results

  • The results tables report JSD and MCon-MAE summaries across five evaluation seeds, with Beam3D and metamaterial comparisons covering their stated metrics and mode counts.Table 11 covers Beam3D MCon-MAE, JSD, and rejection rate; Table 12 compares Bi-FORK and an affine baseline across mode counts; Table 13 reports relaxed MCon-MAE and three JSD metrics.Source: §E.1 · p. 24, Table 11 · p. 25, Table 12 · p. 25

Beam3D

  • For metamaterials, Bi-FORK achieves JSD below the expected finite-sample uniform reference; particle guidance can promote more balanced mode frequencies through dependent sampling.This reference is not a strict lower bound, and full coverage requires a valid sample for every mode.Source: Beam3D · p. 25

E.2 Ablations

The ablations assess bifurcation-mask prediction and generation time, while qualitative figures compare predicted trajectories and modes with ground truth. The figures cover beam evolution and one-, two-, and four-mode samples.

E.2 Ablations

  • Bifurcation-mask prediction is assessed on Beam3D and mechanical metamaterials using accuracy, precision, recall, F1, and balanced accuracy; Allen-Cahn instead receives guidance at every valid frame.Table 14 reports classification on the complete test splits, with mean and standard deviation over five seeds.Source: §E.2 · p. 26, Table 14 · p. 26
  • Beam3D generation time is analyzed for 180 samples over four random seeds, with experiments also described as one input condition over four random draws on a B300 SXM6 with 288 GB.The resulting times are summarized in Table 15; no time values are supplied here.Source: Table 15 · p. 26, §E.2 · p. 26

E.3 Qualitative Results

  • Beam bending figures compare predicted and actual trajectory evolution for 180 samples across node counts from 3 to 11.Predicted trajectories are green and actual trajectories black.Source: Figure 12 · p. 27
  • A sample comparison varies µ and ϵ to show solution diversity alongside the ground truth.Source: Figure 13 · p. 27
  • Bi-FORK visualizations with K = 16 compare ground truth against the closest predicted mode for one-, two-, and four-mode samples.Source: Figure 14 · p. 28, Figure 15 · p. 29, Figure 16 · p. 30
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