Multimodal Neural Operator (MNO)

• Multimodal Neural Operator (MNO) is an operator-learning framework that jointly integrates heterogeneous inputs to learn mappings between function spaces for parametric nonlinear BVPs.
• It leverages a hierarchical architecture inspired by the Fast Multipole Method, achieving parameter efficiency and robust performance on both linear and nonlinear PDEs.
• Its multimodal fusion mechanism experimentally outperforms unimodal baselines, paving the way for future research in higher dimensions and complex geometries.

Searching arXiv for the cited MNO-related papers and terminology. Search query: "(Madala et al., 16 Jul 2025) Multimodal Neural Operator MNO" Multimodal neural operator (MNO) denotes an operator-learning framework in which multiple heterogeneous inputs jointly condition a learned map between function spaces. In its most specific use, the term refers to the architecture introduced in “MNO : A Multi-modal Neural Operator for Parametric Nonlinear BVPs,” where the learned operator maps PDE coefficients, right-hand sides, and boundary data to the solution of a parametric nonlinear boundary value problem in a unified model (Madala et al., 16 Jul 2025). More broadly, the term has been used for operator learners that combine stimulus functions with structured latent descriptors, as in NOBLE for neuron dynamics (Ghafourpour et al., 5 Jun 2025). The acronym is also overloaded in adjacent theory papers, where “MNO” denotes “Multiple Neural Operators” or “Multiple Nonlinear Operator network” rather than “multimodal”; those works address related questions of conditioning a single operator learner on descriptors of operator families (Weihs et al., 29 Oct 2025, Weihs et al., 2 Apr 2026, Weihs et al., 21 May 2026).

1. Formal definition and problem class

In the boundary-value setting, MNO is formulated for parametric nonlinear BVPs on a domain $\Omega \subset \mathbb{R}$, with $\partial \Omega = \{0,1\}$ in the reported 1D experiments. The governing equations are written as

$$A(u;\theta_a)=f \text{ in } \Omega,\qquad B(u;\theta_b)=g \text{ on } \partial\Omega,$$

and instantiated as

$$D_1(x,u;a)=c \text{ for } x\in\Omega,\qquad D_2(x,u;f)=g \text{ for } x\in\partial\Omega.$$

The learned solution operator is

$G:(\theta_a,f,g,\theta_b)\mapsto u,$

while the specific multi-parameter operator learned by the architecture is

$A^\dagger:(a,c,f,g)\mapsto u.$

The central claim is that coefficients such as $a$ parametrize operators acting on $u$, whereas $c$ and $g$ parametrize the range; this modality mismatch is one of the stated reasons why standard unimodal neural operators become difficult to extend to simultaneous variation in coefficients, sources, and boundary terms (Madala et al., 16 Jul 2025).

The reported examples cover three increasingly structured settings. The first is the linear 1D Poisson problem with constant coefficients and Dirichlet boundary conditions, in which the map is $\partial \Omega = \{0,1\}$0. The second is linear 1D Darcy flow,

$\partial \Omega = \{0,1\}$1

with the learned map $\partial \Omega = \{0,1\}$2. The third is a nonlinear first-order BVP with an integral boundary condition,

$\partial \Omega = \{0,1\}$3

with the full multimodal map $\partial \Omega = \{0,1\}$4 (Madala et al., 16 Jul 2025).

A closely related but domain-specific definition appears in NOBLE, where a multimodal operator maps a stimulus function and modality descriptors of a neuron to a voltage trajectory. There the operator is written as

$\partial \Omega = \{0,1\}$5

with $\partial \Omega = \{0,1\}$6 a biologically informed latent vector and $\partial \Omega = \{0,1\}$7 a current-injection time series (Ghafourpour et al., 5 Jun 2025). This broader usage preserves the same structural idea: multiple modalities are lifted into a common operator model rather than treated as isolated regression inputs.

2. Hierarchical architecture for parametric BVPs

The MNO architecture for BVPs is built in three stages: a Generalized FMM (GFMM) block, a Unimodal Neural Operator (UNO) formed by stacking GFMM blocks, and a multimodal fusion mechanism that couples a coefficient branch to an RHS branch across the hierarchy (Madala et al., 16 Jul 2025). The construction is motivated by the Fast Multipole Method, specifically by the rank structure of discretized PDE operators and the corresponding upward and downward signal flow.

In the paper’s 1D GFMM, an input $\partial \Omega = \{0,1\}$8 is partitioned into $\partial \Omega = \{0,1\}$9 blocks of size $$A(u;\theta_a)=f \text{ in } \Omega,\qquad B(u;\theta_b)=g \text{ on } \partial\Omega,$$0. The block uses hierarchical encoder and decoder layers with bridge operators at each level. The encoder computes

$$A(u;\theta_a)=f \text{ in } \Omega,\qquad B(u;\theta_b)=g \text{ on } \partial\Omega,$$1

the top bridge applies

$$A(u;\theta_a)=f \text{ in } \Omega,\qquad B(u;\theta_b)=g \text{ on } \partial\Omega,$$2

and the decoder propagates

$$A(u;\theta_a)=f \text{ in } \Omega,\qquad B(u;\theta_b)=g \text{ on } \partial\Omega,$$3

down to $$A(u;\theta_a)=f \text{ in } \Omega,\qquad B(u;\theta_b)=g \text{ on } \partial\Omega,$$4, with output $$A(u;\theta_a)=f \text{ in } \Omega,\qquad B(u;\theta_b)=g \text{ on } \partial\Omega,$$5. Near-field interactions are carried by the bridge operators $$A(u;\theta_a)=f \text{ in } \Omega,\qquad B(u;\theta_b)=g \text{ on } \partial\Omega,$$6, which are implemented as banded or blocked operators, while far-field interactions are handled by the encoder and decoder blocks $$A(u;\theta_a)=f \text{ in } \Omega,\qquad B(u;\theta_b)=g \text{ on } \partial\Omega,$$7 and $$A(u;\theta_a)=f \text{ in } \Omega,\qquad B(u;\theta_b)=g \text{ on } \partial\Omega,$$8 through hierarchical aggregation and translation (Madala et al., 16 Jul 2025).

The reported parameter count for a linear 1D GFMM block is

$$A(u;\theta_a)=f \text{ in } \Omega,\qquad B(u;\theta_b)=g \text{ on } \partial\Omega,$$9

in contrast to a dense layer with

$$D_1(x,u;a)=c \text{ for } x\in\Omega,\qquad D_2(x,u;f)=g \text{ for } x\in\partial\Omega.$$0

The paper presents this as the main source of parameter efficiency. UNO then stacks multiple GFMM blocks, with a multi-channel extension in which a basis transform becomes a $$D_1(x,u;a)=c \text{ for } x\in\Omega,\qquad D_2(x,u;f)=g \text{ for } x\in\partial\Omega.$$1-D tensor $$D_1(x,u;a)=c \text{ for } x\in\Omega,\qquad D_2(x,u;f)=g \text{ for } x\in\partial\Omega.$$2 acting by channel summation. In that form UNO learns single-parameter maps such as $$D_1(x,u;a)=c \text{ for } x\in\Omega,\qquad D_2(x,u;f)=g \text{ for } x\in\partial\Omega.$$3 or $$D_1(x,u;a)=c \text{ for } x\in\Omega,\qquad D_2(x,u;f)=g \text{ for } x\in\partial\Omega.$$4 (Madala et al., 16 Jul 2025).

MNO extends UNO by using two branches. The coefficient branch ingests modalities such as $$D_1(x,u;a)=c \text{ for } x\in\Omega,\qquad D_2(x,u;f)=g \text{ for } x\in\partial\Omega.$$5, $$D_1(x,u;a)=c \text{ for } x\in\Omega,\qquad D_2(x,u;f)=g \text{ for } x\in\partial\Omega.$$6, and $$D_1(x,u;a)=c \text{ for } x\in\Omega,\qquad D_2(x,u;f)=g \text{ for } x\in\partial\Omega.$$7, and produces latent basis representations at each hierarchical level. The RHS branch ingests $$D_1(x,u;a)=c \text{ for } x\in\Omega,\qquad D_2(x,u;f)=g \text{ for } x\in\partial\Omega.$$8 and $$D_1(x,u;a)=c \text{ for } x\in\Omega,\qquad D_2(x,u;f)=g \text{ for } x\in\partial\Omega.$$9 and maps to $G:(\theta_a,f,g,\theta_b)\mapsto u,$0. The defining multimodal step is an additive fusion in which coefficient-dependent latent corrections perturb the RHS-branch weights before each basis transform:

$G:(\theta_a,f,g,\theta_b)\mapsto u,$1

where $G:(\theta_a,f,g,\theta_b)\mapsto u,$2 is produced by the coefficient branch at the aligned level. The implemented fusion is therefore additive gating via $G:(\theta_a,f,g,\theta_b)\mapsto u,$3 rather than cross-attention or tensor fusion (Madala et al., 16 Jul 2025).

3. Optimization, discretization, and implementation

Training in the BVP paper is purely data-driven and supervised. The loss is mean squared error on the predicted solution,

$G:(\theta_a,f,g,\theta_b)\mapsto u,$4

optimized with Adam. The initial learning rate is $G:(\theta_a,f,g,\theta_b)\mapsto u,$5, and in the Poisson experiment it is decayed to $G:(\theta_a,f,g,\theta_b)\mapsto u,$6 after $G:(\theta_a,f,g,\theta_b)\mapsto u,$7 iterations. Reported evaluation metrics are the backward error

$G:(\theta_a,f,g,\theta_b)\mapsto u,$8

the residual error $G:(\theta_a,f,g,\theta_b)\mapsto u,$9, and the relative error

$A^\dagger:(a,c,f,g)\mapsto u.$0

No physics-informed residual terms are added to the loss (Madala et al., 16 Jul 2025).

Synthetic data are generated by “solution sampling.” The solution is drawn as

$A^\dagger:(a,c,f,g)\mapsto u.$1

where $A^\dagger:(a,c,f,g)\mapsto u.$2 are Chebyshev polynomials and $A^\dagger:(a,c,f,g)\mapsto u.$3; coefficients are then sampled per problem, and $A^\dagger:(a,c,f,g)\mapsto u.$4 and $A^\dagger:(a,c,f,g)\mapsto u.$5 are generated by finite differences applied to the operators $A^\dagger:(a,c,f,g)\mapsto u.$6 and $A^\dagger:(a,c,f,g)\mapsto u.$7. The coefficients $A^\dagger:(a,c,f,g)\mapsto u.$8 are resampled every iteration, which the paper describes as regularization against overfitting. In 1D, experiments use a uniform grid of $A^\dagger:(a,c,f,g)\mapsto u.$9 points on $a$0 and the basis $a$1. The appendix reports preliminary 2D GFMM results on a $a$2 grid using a tensor-product quadtree with Morton ordering, block size $a$3, and four encoder-decoder levels, but the paper explicitly does not claim general resolution-agnostic behavior (Madala et al., 16 Jul 2025).

The default 1D hierarchy uses depth $a$4 and block size $a$5, with block-banded bridges such as tri-diagonal bridges. In the implemented multimodal configuration, the coefficient branch receives $a$6 as channels and the RHS branch receives $a$7. The nonlinear GFMM uses the rational activation

$a$8

which the paper motivates by the observation that inverse operators for linear PDEs depend rationally on coefficients and reports as empirically superior to ReLU for multimodal tasks. Training uses single precision on two NVIDIA TITAN RTX GPUs (Madala et al., 16 Jul 2025).

4. Empirical behavior on linear and nonlinear boundary value problems

In the discrete 1D Poisson experiment, the paper compares UNO, FNO, and DeepONet at similar parameter scales. UNO, configured as two linear GFMM blocks with $a$9 and $u$0, has $u$1 parameters; the reported FNO has $u$2 parameters, and DeepONet has $u$3. Under solution sampling with in-distribution $u$4, the reported errors are $u$5 and $u$6 for FNO, $u$7 and $u$8 for DeepONet, and $u$9 and $c$0 for UNO. On OOD solution sampling with $c$1, UNO remains at $c$2 and $c$3, whereas FNO and DeepONet degrade more strongly. Under RHS-sampled OOD, all models deteriorate, but UNO still yields lower backward error than the baselines (Madala et al., 16 Jul 2025).

For 1D Darcy flow, the comparison is between multimodal MNO and several unimodal UNOs that receive only $c$4 while being trained under fixed or mixed coefficient distributions. On tests with quadratic coefficients $c$5, MNO reports $c$6 and $c$7, compared with $c$8 and $c$9 for UNO-aQ, and markedly larger errors for UNO-aLN and UNO-mix. On tests with log-normal coefficients $g$0, MNO reports $g$1 and $g$2, while the unimodal baselines show much larger relative errors, including $g$3 for UNO-aQ and $g$4 for UNO-aLN. The stated interpretation is that unimodal training cannot cope with coefficient shifts, whereas MNO learns $g$5 across heterogeneous $g$6 distributions (Madala et al., 16 Jul 2025).

The nonlinear BVP provides the most complete multimodal test. When $g$7 and $g$8, UNO reports $g$9, $\partial \Omega = \{0,1\}$00, and $\partial \Omega = \{0,1\}$01, whereas MNO reports $\partial \Omega = \{0,1\}$02, $\partial \Omega = \{0,1\}$03, and $\partial \Omega = \{0,1\}$04. Under simultaneous OOD shifts in both $\partial \Omega = \{0,1\}$05 and $\partial \Omega = \{0,1\}$06, UNO degrades to $\partial \Omega = \{0,1\}$07, $\partial \Omega = \{0,1\}$08, and $\partial \Omega = \{0,1\}$09, while MNO remains at $\partial \Omega = \{0,1\}$10, $\partial \Omega = \{0,1\}$11, and $\partial \Omega = \{0,1\}$12 (Madala et al., 16 Jul 2025).

The paper also isolates the gap between solution-sampled and RHS-sampled OOD regimes. For Poisson, residual errors rise from $\partial \Omega = \{0,1\}$13 to $\partial \Omega = \{0,1\}$14 for UNO; for Darcy, from $\partial \Omega = \{0,1\}$15 to $\partial \Omega = \{0,1\}$16 for MNO; and for the nonlinear BVP, from $\partial \Omega = \{0,1\}$17 to $\partial \Omega = \{0,1\}$18 for MNO. This indicates that multimodal conditioning improves robustness, but does not remove the difficulty of RHS-sampled OOD generalization (Madala et al., 16 Jul 2025).

5. Broader multimodal operator formulations

The broader concept of multimodal operator learning predates the specific GFMM-based MNO. MIONet studies operators defined on a product of Banach spaces,

$\partial \Omega = \{0,1\}$19

and proves a universal approximation theorem for continuous multiple-input operators on compact product sets. Its low-rank realization uses several branch nets, one per modality, and a trunk net for the output domain, with the forward map written as an elementwise tensor-product factorization of branch features and trunk features. In the low-rank form, the approximation is

$\partial \Omega = \{0,1\}$20

This establishes a direct antecedent for multimodal operator learning in the sense of multiple functional inputs, although the later GFMM-based MNO uses a different hierarchical mechanism and a different fusion rule (Jin et al., 2022).

NOBLE provides a second, domain-specific instantiation of MNO in computational neuroscience. It conditions a temporal FNO on a biologically informed latent vector $\partial \Omega = \{0,1\}$21 together with a current-injection time series $\partial \Omega = \{0,1\}$22, after converting both to frequency-modulated, time-aligned sinusoidal embeddings. The operator learner then predicts somatic voltage trajectories $\partial \Omega = \{0,1\}$23. In the reported configuration, NOBLE uses a 1D FNO with 12 layers, 24 hidden channels, and 256 retained Fourier modes, for approximately $\partial \Omega = \{0,1\}$24 parameters. It is trained on $\partial \Omega = \{0,1\}$25 samples generated from 60 hall-of-fame PVALB biophysical models built on NEURON using the BioNet framework, with 50 models for training and 10 for testing. Reported performance includes a test $\partial \Omega = \{0,1\}$26 relative error of $\partial \Omega = \{0,1\}$27 on held-out current injections for the 50 training HoF models and a measured $\partial \Omega = \{0,1\}$28 speedup over the numerical solver at batch size 1000. Morphology and transcriptomics are used upstream in data generation and validation, but in this instantiation they are not direct operator inputs (Ghafourpour et al., 5 Jun 2025).

These two lines of work show that “multimodal” can refer either to jointly varying PDE coefficients, sources, and boundary operators, or to stimulus functions conditioned on interpretable latent descriptors. In both cases, the operative distinction from standard unimodal operator learning is that the learned map is defined on a product of heterogeneous modalities rather than on a single functional argument.

6. Theoretical and terminological context

A terminological complication is that several recent theory papers use the acronym “MNO” differently. “A Deep Learning Framework for Multi-Operator Learning: Architectures and Approximation Theory” defines MNO as a “Multiple Nonlinear Operator network,” with the separable form

$\partial \Omega = \{0,1\}$29

where $\partial \Omega = \{0,1\}$30 is an operator descriptor, $\partial \Omega = \{0,1\}$31 is an input function, and $\partial \Omega = \{0,1\}$32 is a query location. That work distinguishes multiple operator learning from learning several distinct single operators, establishes universal approximation results in continuous and measurable settings, and derives worst-case Lipschitz scaling laws for approximation complexity (Weihs et al., 29 Oct 2025).

The later papers “Generalization Bounds and Statistical Guarantees for Multi-Task and Multiple Operator Learning with MNO Networks” and “Multiple Neural Operators Achieve Near-Optimal Rates for Multi-Task Learning” analyze related separable architectures under hierarchical sampling of triples $\partial \Omega = \{0,1\}$33. Their MNO processes these three modalities through distinct subnetworks and recombines them multiplicatively, yielding explicit approximation-estimation tradeoffs and learning-rate statements in the operator-sampling budget $\partial \Omega = \{0,1\}$34. One reported rate is

$\partial \Omega = \{0,1\}$35

while the later near-optimality paper states that shared representations across tasks do not increase the overall cost and that multi-task operator learning follows the same scaling laws as single operator learning up to logarithmic factors (Weihs et al., 2 Apr 2026, Weihs et al., 21 May 2026).

This theoretical literature is not about the GFMM-based multimodal BVP architecture specifically. It is instead about separable multi-operator conditioning, where $\partial \Omega = \{0,1\}$36 plays the role of an operator or task descriptor. The overlap is conceptual rather than architectural. A plausible implication is that the multimodal BVP MNO can be viewed as one concrete engineering realization of the broader problem of learning conditioned operator families, but its additive hierarchical fusion and rational activations are not the objects analyzed in those generalization papers.

7. Limitations and open questions

The BVP MNO paper states several limitations directly. The demonstrations are primarily 1D, with only preliminary 2D GFMM results in the appendix; future work is identified as higher dimensions, time-dependent PDEs, and diverse geometries. Generalization to RHS-sampled OOD remains challenging, and the paper notes that blending with iterative solvers can improve robustness. It also states that no formal approximation or stability theorems for MNO are proved, and that more rigorous analysis of activation choices and theoretical guarantees is planned (Madala et al., 16 Jul 2025).

NOBLE is limited in scope to somatic voltage responses to current injections for a single PVALB neuron’s population of hall-of-fame models. Its multimodal conditioning is also narrower than the label may suggest: the direct inputs are electrophysiological latent features and stimulus only, while morphology and transcriptomics are not yet embedded as model inputs. The paper leaves handling missing modalities, synaptic input conductances, multi-neuron interactions, dendritic recordings, and extrapolation outside the convex hull of training latent features for future work (Ghafourpour et al., 5 Jun 2025).

The theory papers impose bounded-domain, Lipschitz, separability, clipping, and independence assumptions, and they explicitly identify the extension to non-separable architectures such as attention-based multimodal operators as an open direction (Weihs et al., 2 Apr 2026). Taken together, these limitations mark a current division in the literature: multimodal operator architectures have advanced empirically in PDEs and biological dynamics, while rigorous guarantees are stronger for separable multiple-operator models than for the fast-multipole-inspired fusion architectures now used in practice.