differentiable programming of reaction-diffusion patterns
TRANSCRIPT
Differentiable Programming of Reaction-Diffusion Patterns
Alexander Mordvintsev1, Ettore Randazzo1 and Eyvind Niklasson1
Abstract
Reaction-Diffusion (RD) systems provide a computationalframework that governs many pattern formation processes innature. Current RD system design practices boil down totrial-and-error parameter search. We propose a differentiableoptimization method for learning the RD system parametersto perform example-based texture synthesis on a 2D plane.We do this by representing the RD system as a variant ofNeural Cellular Automata and using task-specific differen-tiable loss functions. RD systems generated by our methodexhibit robust, non-trivial “life-like” behavior.
IntroductionMulticellular organisms build and maintain their body struc-tures through the distributed process of local interactionsamong tiny cells working toward a common global goal.This approach, often called self-organisation, is drasticallydifferent from the way most human technologies work. Weare just beginning to scratch the surface of integrating someof nature’s “best practices” into technology.
In 1952, Alan Turin wrote his seminal work, “The Chem-ical Basis of Morphogenesis” (Turing, 1952), in which heproposed that pattern formation in living organisms can becontrolled by systems of chemical substances called mor-phogens. Simultaneous processes of chemical Reactions-Diffusion (RD) of these substances provide the sufficientbiocomputation platform to execute distributed algorithmsof pattern formation (Kondo and Miura, 2010; Landge et al.,2019).
Even a system of just two interacting substances (e.g.Gray-Scott) can produce a great diversity of patterns and in-teresting behaviours (Pearson, 1993). As is often the casewith complex emergent behaviours of simple systems, it isvery difficult find model parameters that produce a particu-lar, predefined behaviour. Most of the time researchers usehand-tuned reaction rules, random and grid search over theparameter space for the combinations with the desired prop-erties. Procedural texture synthesis is one of the best knownapplications for this type of parameter tuning (Witkin andKass, 1991; Turk, 1991). In this paper, we propose to use
differentiable optimization as an alternative to such a trial-and-error process.
The task of determining the sets of parameters that leadto desired behaviors becomes even more important as weenter the realm of artificial life and synthetic biology. Thework of Scalise and Schulman (2014) is a remarkable exam-ple of an attempt to design a flexible modular framework forRD-based spatial programming. The authors demonstrate(at least in simulation) a way to combine a set of basic com-putational primitives, implemented as RD systems of DNAstrands, into multistage programs that generate non-trivialspatial structures. We argue that these programs cannot yetbe called “self-organizing systems” due to two importantlimitations. First, they rely on a predefined initial state thatdefines the global coordinate system on which the programoperates through chemical gradients or precise locations ofchemical “seeds”. Second, program execution is a feed-forward process that does not imply homeostatic feedbackloops that make the patterns robust to external perturbationsor a changed initial state.
Another very related line of research in artificial life isLenia (Chan, 2020), which aims to find rules and config-urations for space-time-state-continuous cellular automatathat demonstrate life-like homeostatic and replicating be-haviours.
Figures in video form and a reference implementation isavailable here1.
Differentiable Reaction-Diffusion ModelWe study a computational model that can be defined by thefollowing system of PDEs:
∂xi∂t
= ci∇2xi + fθ(x0, . . . , xn−1)
x0, . . . , xn−1 are scalar fields representing the “concentra-tions” n “chemicals” on a 2D plane. ci are per-”chemical”diffusion coefficients, and fθ : Rn → Rn is a function defin-ing the rate of change of “chemical concentrations” due to
1https://selforglive.github.io/alife_rd_textures/
arX
iv:2
107.
0686
2v1
[cs
.NE
] 2
2 Ju
n 20
21
local “reactions”. Chemical terms are used here in quotesbecause, in the current version of our model, the function fθneed not obey any actual physical laws, such as the law ofconservation of mass or the law of mass action. “Concentra-tions” can also become negative.
Reaction-Diffusion as a Neural CAThe objective of this paper is to train a model that can be rep-resented by a space-time continuous PDE system. Yet, wehave to discretize the model to run it on a digital computer.The discretized model can be treated as a case of NeuralCellular Automata (NCA) model (Mordvintsev et al., 2020;Randazzo et al., 2020).
Our model and the training procedure are heavily inspiredby the texture-synthesizing Neural CA described by Niklas-son et al. (2021). Similarly, we discretize space into cellsand time into steps, use explicit Euler integration of thesystem dynamics and backpropagation-through-time to op-timize the model parameters. Our model differs from theprevious texture-synthesizing NCA:
• CA iteration does not have the perception phase. The “re-action” part of cell update (modelled by fθ) depends onthe current state of the cell only.
• There is an isotropic diffusion process that runs in paral-lel with “reaction” and is modelled by channelwise con-volution with a Laplacian kernel. This is the only methodof communication between cells. Thus, the Neural RDmodel is completely isotropic: in addition to translation,the system behaviour is invariant to rotational and mirror-ing transformations.
• We do not use stochastic updates, all cells are updated atevery step.
Thus, the system described here can be seen as a strippedversion of the Neural CA model. These restrictions are moti-vated by a number of practical and philosophical argumentsdescribed below.
Model simplicity and prospects of physical implementa-tion The discussion section of “Growing NCA” article byMordvintsev et al. (2020) contains speculations about poten-tial of physical implementation of the Neural CA as a gridof tiny independent computers (microcontrollers). NeuralCA consists of a grid of discrete cells that are sophisticatedenough to persist and update individual state vectors, andcommunicate within the neighborhood in a way that differ-entiates between neighbours, adjacent to different sides ofthe cell. This implies the existence of global alignment be-tween cells, so that they agree where up and left are, andseparate communication channels to send the informationin different directions. In contrast, diffusion based com-munication doesn’t require any dedicated structures, and
“just happends” in nature due to the Brownian motion ofmolecules.
Furthermore, states of Neural CA cells are only modifiedby their learned update rules and are not affected by any en-vironmental processes. Cells have clear separation betweenwhat’s inside and outside of them, and can control whichsignals should be let through. However, many natural phe-nomena of self-organisation can be effectively described asa PDE system on a continuous domain. Individual elementsthat constitute the domain are considered negligibly small,and the notion of their individual updates is meaningless.
In the experiments section we cover some practical ad-vantages of the proposed RD model with respect to NeuralCA. In particular we demonstrate the generalization into ar-bitrary mesh surfaces and even volumes.
Reaction-Diffusion CA update rule The discrete updaterule can be written as follows:
xt+1i = xti + ciKlap ∗ xti d+ fθ(x
t0, . . . , x
tn−1) r
d = ∆t/∆2h, r = ∆t
(1)
Klap is a 3x3 Laplacian convolution kernel, ci and θ areparameters that control the CA behaviour, and the coeffi-cients r and d control the rates of reaction and diffusion,encapsulating temporal and spatial discretization step sizes∆t and ∆h. By varying these coefficients we can validateif the learned discrete CA rule approximates the continuousPDE and does not over-feat to the particular discretization.During training we use r = d = 1.0.
The function fθ(x) = act(xW0 + b0)W1 is a smalltwo layer neural network with parameters θ : (W0 ∈Rn×h,W1 ∈ Rh×n, b0 ∈ Rh) and a non-linear element-wise activation function act (see the experiments section).In our experiments we use n = 32, h = 128, so the systemmodels the dynamics of 32 “chemicals”, and the total num-ber of network parameters equals to 8320. Per-“chemical”diffusion coefficients ci can be learned (in this case we setci = sigmoid(ci) to make sure that diffusion rate stays in0..1 range), or fixed to specific values.
Texture synthesisReaction-Diffusion models are a well-known tool for tex-ture synthesis. Typically, manual parameter tuning has beenused to design RD systems that generate desired visual pat-terns (Witkin and Kass, 1991; Turk, 1991). We propose anexample-based training procedure to learn a RD rule for agiven example texture image. This procedure closely fol-lows the work “Self-Organising Textures” (SOT) (Niklassonet al., 2021), with a few important modifications. Our goalis to learn a RD update function whose continuous applica-tion from a starting state would produce a pattern similar tothe provided texture sample. The procedure is summarisedin algorithm 1.
Algorithm 1: Texture RD system training
pool← Npool randomly sampled seed states ;for step← 1 to Ntrain do
batch← sample Nbatch random pool indices ;x← pool[batch];if step mod Rseed = 0 then
x[0]← random seed stateendfor i← 1 to Uniform(Imin, Imax) do
x← RDθ(x) ;endloss← calcLoss(x) ;update θ using the gradient of loss ;pool[batch]←x;
endNtrain 20000 Imin 32 Npool 1024Rseed 32 Imax 96 Nbatch 4
Seed states SOT uses seed states filled with zero values.Stochastic cell updates provide the sufficient variance tobreak the symmetry between cells. We use synchronous ex-plicit Euler integration scheme that updates all cells simulta-neously, so the non-uniformity of the seed states is requiredto break the symmetry. We initialize the grid with a numberof sparsely scattered Gaussian blobs (fig.1).
Figure 1: Random seed states.
Periodic injection of seed states into training batches iscrucial to prevent the model from forgetting how to developthe pattern from the seed, rather then improving already ex-isting one. We observed that it is sufficient to inject the seedmuch less often than in SOT. We use Rseed = 32 in thisexperiment.
Rotation-invariant texture loss Similar to SOT, we in-terpret the first 3 “chemicals” as RGB colors and use themto calculate texture loss. Our texture loss is also based onmatching Gram matrices of pre-trained VGG network ac-tivations (Gatys et al., 2015), but has an important modifi-cation to account for the rotational invariance of isotropicRD. Consider the texture lined 0118 from the figure 2.The target pattern is anisotropic, but RD, unlike NCA, hasno directional information and cannot produce a texture thatexactly matches the target orientation. We construct therotation-invariant texture loss by uniformly sampling Nrot
target images rotated by angles 0◦...360◦ and computing thecorresponding texture descriptions. This computation onlyoccurs in the initialization phase and does not slow downthe training. At each loss evaluation, the texture descriptorsfrom RD are matched with all target orientations and the bestmatch is taken for each sample in the batch.
Per-“chemical” diffusion coefficients in Eq. (1) are set toc0..7 = 1/8, c8..15 = 1/4, c16..23 = 1/2, c24..31 = 1, so thatthe “substances” are split into 4 groups of varying diffusiv-ity. Therefore, RGB colors correspond to slowly diffusingchannels x0..2. We experimented with learned diffusion co-efficients, but it did not seem to bring substantial improve-ment, so we kept fixed values for the simplicity. In all of thetexture synthesis experiments we use wrap-around (torus)grid boundary conditions.
RD network uses a variant of Swish (Ramachandran et al.,2017) elementwise activation function: act(x) = xσ(5.0 ∗x), where σ is a sigmoid function.
We trained seven texture-synthesizing RD models (fig.2).Six were using 128x128 images from DTD dataset (Cimpoiet al., 2014), and the last was using a 48x48 lizard emojiimage, replicated four times. Models were trained for 20000steps using the Adam optimiser (Kingma and Ba, 2015) withlearning rate 1e-3 decayed by 0.2 at steps 1000 and 10000.We also used the gradient normalization trick from SOT toimprove training stability. Training a single RD model tookabout 1 hour on the NVIDIA P100 GPU.
ResultsIn spite of the constrained computational model, trained RDsystems were capable of reproducing (although imperfectly)distinctive features of the target textures. All models exceptchequered 0121 seemed to be isotropic, which mani-fested in the random orientation of the resulting patterns, de-pending on the randomized seed state. chequered 0121always produced diamond-oriented squares, which suggestsoverfitting to the particular discrete grid. Below we investi-gate this behaviour more carefully.
Witkin and Kass (1991) proposed using anisotropic diffu-sion for anisotropic texture generation with RD. In our ex-periments, we demonstrated the capability of fully isotropicRD systems to produce locally anisotropic textures throughthe process of iterative alignment, that looks surprisingly“life-like”. Figure 3 shows snapshots of grid states at dif-ferent stages of pattern evolution. We recommend watchingsupplementary videos to get a better intuition of RD systembehaviours.
Do we really learn a PDE? We decided to validate thatthe discrete Neural CA that we use to simulate RD systemis robust enough to be used with different grid resolutionthan used during training. This would confirm that the sys-tem we trained really approximates a space-time continuous
Figure 2: Left column: sample texture images; othercolumns: different 5000-step runs of the learned RD mod-els. Almost all models replicate features of the target tex-ture in rotation-invariant fashion. Only chequered 0121overfitted to exploit the underlying raster grid structure toalways produce diamond-oriented checker squares.
Figure 3: Stages of texture development at time steps 50,100, 1000 and 10000. Anisotropic textures form by localsymmetry breaking and refinement. Resulting patterns lookconsistent over long time periods, but never reach full sta-bility. For example, dots in the middle row asynchronouslychange colors.
Figure 4: Running RD system on a non-uniform r grid. Allmodels except the rightmost seem to be capable of oper-ating on a modified grid, preserving the key pattern char-acteristics. chequered 0121 overfitted to the particulargrid resolution and is unable to produce right corners atlarger scales. Sometimes it even develops instabilities andexplodes.
PDE without overfitting to the particular grid discretization.One way to execute the RD on a finer grid is to decrease the∆h coefficient in (1). This leads to the quick growth of thediffusion term, making the simulation unstable. We can mit-igate this by reducing the ∆t as well. In practice we keepd = 1.0, but decrease r, so that ∆h =
√r. Thus, setting
r = 1/4 corresponds to running on a twice finer grid, andshould produce patterns magnified two times. Decrease of rmay also be interpreted as reaction speed slowdown or dif-fusion acceleration.
Fig. 4 shows results of RD system evaluation on a gridhaving non-uniform r: in the center r = 1, slowly decreas-ing to r ≈ 1/9 at the boundary. Most of the trained modelswere capable of preserving their behaviour independent ofthe grid resolution. chequered 0121 model was the only
Figure 5: RD texture models can easily be applied to meshsurfaces. We treat each vertex of the Stanford Bunny modelas a cell, and allow the associated state vectors to diffusealong the mesh edges. This enables consistent texturing ofdense meshes without constructing UV-maps or tangent co-ordinate frames. Please see the supplementary videos for thesystem dynamics and more views.
exception. The grid-overfitting hypothesis was confirmed bythe fact that the model could not form right angles at the gridcorners, and even developed instabilities in the fine resolu-tion grid areas.
Generalization beyond 2D plane RD-system can be ap-plied to any manifold that has a diffusion operation definedon it. This enables much more extreme out-of-training gen-eralization scenarios, than those possible for Neural CA. Forexample, applying texture synthesis method by Niklassonet al. (2021) to a 3d mesh surface would require definingsmooth local tangent coordinate systems for all surface cells(for example, located at mesh vertices). This is necessaryto compute partial derivatives of cell states with some vari-ant of generalized of Sobel filters. In contrast, Neural RDdoesn’t require tangents, and can be applied to an arbitrarymesh by state diffusion over the edges of the mesh graph(see fig. 5). Even more surprisingly, RD models that weretrained to produce patterns on a 2d plane, can be applied tospaces of higher dimensionality by just replacing the diffu-sion operator with a 3d equivalent. Figure 6 shows examplesof volumetric texture synthesis by 2d models.
DiscussionReaction-Diffusion is known to be an important mechanismcontrolling many developmental processes in nature (Kondoand Miura, 2010; Landge et al., 2019). We think that master-ing RD-system engineering is an important prerequisite for
Figure 6: 2D to 3D generalization of Reaction-Diffusionmodels. The models that were trained on 2D plane with2D image loss and executed on a 3D space. For somepatterns, individual slices though the volume have texturessimilar to the target images, while for others (e.g. thelast row) the similarity is less convincing. We treat whitecolor as transparent to visualize the structure of lizardsand polka-dotted patterns. Please see supplementaryvideos for model dynamics.
making human technology more life-like: robust, flexible,and sustainable. This work demonstrates the applicability ofDifferentiable Programming to the design of RD-systems.We think this is an important stepping stone to transformRD into a practical engineering tool.
To achieve this, some limitations should be addressed infuture work. First, it is crucial to find such optimizationproblem formulations that would produce physically plau-sible RD systems. Second, further research in the area ofdifferentiable objective formulations is needed to make thisapproach applicable to a broader range of design problemsfor self-organizing systems.
ReferencesChan, B. (2020). Lenia and expanded universe. ArXiv,
abs/2005.03742.
Cimpoi, M., Maji, S., Kokkinos, I., Mohamed, S., and Vedaldi, A.(2014). Describing textures in the wild. 2014 IEEE Con-ference on Computer Vision and Pattern Recognition, pages3606–3613.
Gatys, L. A., Ecker, A. S., and Bethge, M. (2015). Texture synthe-sis using convolutional neural networks. In NIPS.
Kingma, D. P. and Ba, J. (2015). Adam: A method for stochasticoptimization. CoRR, abs/1412.6980.
Kondo, S. and Miura, T. (2010). Reaction-diffusion model as aframework for understanding biological pattern formation.Science, 329:1616 – 1620.
Landge, A. N., Jordan, B. M., Diego, X., and Muller, P. (2019).Pattern formation mechanisms of self-organizing reaction-diffusion systems. Developmental Biology, 460:2 – 11.
Mordvintsev, A., Randazzo, E., Niklasson, E., and Levin,M. (2020). Growing neural cellular automata. Distill.https://distill.pub/2020/growing-ca.
Niklasson, E., Mordvintsev, A., Randazzo, E., andLevin, M. (2021). Self-organising textures. Distill.https://distill.pub/selforg/2021/textures.
Pearson, J. E. (1993). Complex patterns in a simple system. Sci-ence, 261(5118):189–192.
Ramachandran, P., Zoph, B., and Le, Q. V. (2017). Swish: a self-gated activation function. arXiv: Neural and EvolutionaryComputing.
Randazzo, E., Mordvintsev, A., Niklasson, E., Levin, M., andGreydanus, S. (2020). Self-classifying mnist digits. Distill.https://distill.pub/2020/selforg/mnist.
Scalise, D. and Schulman, R. (2014). Designing modular reaction-diffusion programs for complex pattern formation.
Turing, A. M. (1952). The chemical basis of morphogenesis. Philo-sophical transactions of the Royal Society of London. SeriesB, Biological sciences, 237(641):37–72.
Turk, G. (1991). Generating textures on arbitrary surfaces usingreaction-diffusion. Proceedings of the 18th annual confer-ence on Computer graphics and interactive techniques.
Witkin, A. and Kass, M. (1991). Reaction-diffusion textures. Pro-ceedings of the 18th annual conference on Computer graph-ics and interactive techniques.