2026-09-21
B-BiLO runs HMC on PDE parameters while LoRA-tunes a local neural operator, matching a numerical Poisson posterior and reaching 6.2% relative L2 on 2D Darcy.
Bayesian PDE inverse problems want a posterior over parameters, not a point estimate. Each draw usually means another forward solve, plus a gradient. Adjoints are painful to derive; black-box solvers do not expose them. PINNs dump data and residual into one weighted loss. Bayesian PINNs go further and run HMC on the weights as well, which explodes dimension. Tightening the residual noise σf improves physics and kills step size, so effective sample size per second falls.
Neural operators evaluate quickly after a large synthetic pretraining corpus, and they drift off the training measure. The missing tool is gradient-based sampling that needs no adjoint, no pretraining dataset, and no MCMC in weight space.
This Nature Communications paper is the journal version of arXiv:2507.17019. The method is B-BiLO.
The upper level samples PDE parameters θ with HMC. The lower level does not sample weights. It fits a network as a local solution operator: at the current θ, both the PDE residual and the total derivative of that residual w.r.t. θ should vanish. The second condition keeps nearby solutions on-PDE, so the potential U(θ) can be differentiated through the network without an adjoint.
Weights are pretrained at an initial θ0, optionally with one numerical snapshot as extra data. During sampling, θ moves in small leapfrog steps and the net is fine-tuned from the previous state. Fine-tuning can be full or LoRA: freeze the pretrained weights and learn a low-rank increment AB, denoted LoRA(r). The lower level stops at tolerance ε. Under mild smoothness, both the gradient error and the KL to the ideal posterior are O(ε).
Unknown spatial fields, such as Darcy diffusivity, are handled with an auxiliary variable z=f(x) and a 64-term KL expansion.
Nonlinear Poisson, inferring k (truth 0.7): solutions are odd, so u(0) should carry no uncertainty. B-BiLO's posterior standard deviation at the origin is near zero, and the parameter posterior tracks a Metropolis–Hastings plus numerical-solver reference. BPINN with σf=0.1 still shows uncertainty at the origin; smaller σf buys accuracy and loses ESS/sec. B-BiLO has no σf to tune. LoRA ranks 4 and 8 match full fine-tuning on the k posterior. Peak memory under LoRA stays below full FT; wall time per sample is worse on small nets and better once width grows.
Gene-expression particle model, inferring birth rate λ and degradation μ: LoRA rank 4 is within 2.1% on 90% HDPR area, 2.0% on λ MAP, 4.0% on μ MAP, versus 3.5% / 1.8% / 3.4% for full FT.
2D Darcy: 21×21 observations, 10% noise, 61×61 collocation. LoRA rank 4 posterior mean is 6.2% relative L2 from the numerical reference and recovers high/low diffusivity regions.
2D glioblastoma MRI: Fisher–KPP inference of nondimensional D, ρ, and two density thresholds. LoRA rank 8 posteriors match full FT; MAP segmentations track WT/TC on MRI, with a u=1% infiltration margin slightly beyond the visible lesion. There is no in-vivo density ground truth; 3D cases are left open.
This is a practical compromise for scientific UQ: the forward map is a locally fitted net, and uncertainty lives only in low-dimensional physics parameters. For tumor growth or permeability, LoRA cuts trainable weights on every leapfrog step. It does not replace adjoint FEM. It does replace BPINN-style MCMC over weights.
3D and higher-dimensional fields are explicit future work. LoRA is slower on narrow nets; the win appears at larger width. The lower-level tolerance ε enters the posterior at O(ε), and there is no universal rule for setting it. Darcy is truncated at 64 KL modes. The tumor study is one 2D case. The Nature PDF was behind a login wall; methods and numbers are taken from the authors' contemporaneous preprint 2507.17019v2, which matches the published abstract and author list.